3.5.40 \(\int (d+e x)^m (b x+c x^2)^3 \, dx\) [440]

Optimal. Leaf size=267 \[ \frac {d^3 (c d-b e)^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{2+m}}{e^7 (2+m)}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {3 c^2 (2 c d-b e) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^7 (7+m)} \]

[Out]

d^3*(-b*e+c*d)^3*(e*x+d)^(1+m)/e^7/(1+m)-3*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*(e*x+d)^(2+m)/e^7/(2+m)+3*d*(-b*e+c*d
)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(3+m)/e^7/(3+m)-(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(
4+m)/e^7/(4+m)+3*c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(5+m)/e^7/(5+m)-3*c^2*(-b*e+2*c*d)*(e*x+d)^(6+m)/e^7/
(6+m)+c^3*(e*x+d)^(7+m)/e^7/(7+m)

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Rubi [A]
time = 0.12, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \begin {gather*} \frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}-\frac {3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac {d^3 (c d-b e)^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{m+2}}{e^7 (m+2)}+\frac {c^3 (d+e x)^{m+7}}{e^7 (m+7)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(b*x + c*x^2)^3,x]

[Out]

(d^3*(c*d - b*e)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (3*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(2 + m))/(e
^7*(2 + m)) + (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(3 + m))/(e^7*(3 + m)) - ((2*c*d -
b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^(4 + m))/(e^7*(4 + m)) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*
e^2)*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (3*c^2*(2*c*d - b*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)
^(7 + m))/(e^7*(7 + m))

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx &=\int \left (\frac {d^3 (c d-b e)^3 (d+e x)^m}{e^6}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{1+m}}{e^6}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{2+m}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{3+m}}{e^6}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{4+m}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{5+m}}{e^6}+\frac {c^3 (d+e x)^{6+m}}{e^6}\right ) \, dx\\ &=\frac {d^3 (c d-b e)^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{2+m}}{e^7 (2+m)}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {3 c^2 (2 c d-b e) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^7 (7+m)}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 236, normalized size = 0.88 \begin {gather*} \frac {(d+e x)^{1+m} \left (\frac {d^3 (c d-b e)^3}{1+m}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)}{2+m}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^2}{3+m}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^3}{4+m}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^4}{5+m}-\frac {3 c^2 (2 c d-b e) (d+e x)^5}{6+m}+\frac {c^3 (d+e x)^6}{7+m}\right )}{e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(b*x + c*x^2)^3,x]

[Out]

((d + e*x)^(1 + m)*((d^3*(c*d - b*e)^3)/(1 + m) - (3*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x))/(2 + m) + (3*d
*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2)/(3 + m) - ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e
+ b^2*e^2)*(d + e*x)^3)/(4 + m) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4)/(5 + m) - (3*c^2*(2*c*d
- b*e)*(d + e*x)^5)/(6 + m) + (c^3*(d + e*x)^6)/(7 + m)))/e^7

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(903\) vs. \(2(267)=534\).
time = 0.48, size = 904, normalized size = 3.39 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

c^3/(7+m)*x^7*exp(m*ln(e*x+d))+(b^3*e^3*m^3+3*b^2*c*d*e^2*m^3+18*b^3*e^3*m^2+39*b^2*c*d*e^2*m^2-15*b*c^2*d^2*e
*m^2+107*b^3*e^3*m+126*b^2*c*d*e^2*m-105*b*c^2*d^2*e*m+30*c^3*d^3*m+210*b^3*e^3)/e^3/(m^4+22*m^3+179*m^2+638*m
+840)*x^4*exp(m*ln(e*x+d))+c^2*(3*b*e*m+c*d*m+21*b*e)/e/(m^2+13*m+42)*x^6*exp(m*ln(e*x+d))+m*d*(b^3*e^3*m^3+18
*b^3*e^3*m^2-12*b^2*c*d*e^2*m^2+107*b^3*e^3*m-156*b^2*c*d*e^2*m+60*b*c^2*d^2*e*m+210*b^3*e^3-504*b^2*c*d*e^2+4
20*b*c^2*d^2*e-120*c^3*d^3)/e^4/(m^5+25*m^4+245*m^3+1175*m^2+2754*m+2520)*x^3*exp(m*ln(e*x+d))-6*d^4*(b^3*e^3*
m^3+18*b^3*e^3*m^2-12*b^2*c*d*e^2*m^2+107*b^3*e^3*m-156*b^2*c*d*e^2*m+60*b*c^2*d^2*e*m+210*b^3*e^3-504*b^2*c*d
*e^2+420*b*c^2*d^2*e-120*c^3*d^3)/e^7/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)*exp(m*ln(e
*x+d))+3*(b^2*e^2*m^2+b*c*d*e*m^2+13*b^2*e^2*m+7*b*c*d*e*m-2*c^2*d^2*m+42*b^2*e^2)*c/e^2/(m^3+18*m^2+107*m+210
)*x^5*exp(m*ln(e*x+d))+6/e^6*m*d^3*(b^3*e^3*m^3+18*b^3*e^3*m^2-12*b^2*c*d*e^2*m^2+107*b^3*e^3*m-156*b^2*c*d*e^
2*m+60*b*c^2*d^2*e*m+210*b^3*e^3-504*b^2*c*d*e^2+420*b*c^2*d^2*e-120*c^3*d^3)/(m^7+28*m^6+322*m^5+1960*m^4+676
9*m^3+13132*m^2+13068*m+5040)*x*exp(m*ln(e*x+d))-3*(b^3*e^3*m^3+18*b^3*e^3*m^2-12*b^2*c*d*e^2*m^2+107*b^3*e^3*
m-156*b^2*c*d*e^2*m+60*b*c^2*d^2*e*m+210*b^3*e^3-504*b^2*c*d*e^2+420*b*c^2*d^2*e-120*c^3*d^3)*d^2/e^5*m/(m^6+2
7*m^5+295*m^4+1665*m^3+5104*m^2+8028*m+5040)*x^2*exp(m*ln(e*x+d))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 663 vs. \(2 (273) = 546\).
time = 0.34, size = 663, normalized size = 2.48 \begin {gather*} \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} e^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d x^{3} e^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} x^{2} e^{2} + 6 \, d^{3} m x e - 6 \, d^{4}\right )} b^{3} e^{\left (m \log \left (x e + d\right ) - 4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} e^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d x^{4} e^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} x^{3} e^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} x^{2} e^{2} - 24 \, d^{4} m x e + 24 \, d^{5}\right )} b^{2} c e^{\left (m \log \left (x e + d\right ) - 5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} + \frac {3 \, {\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} x^{6} e^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d x^{5} e^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} x^{4} e^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} x^{3} e^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} x^{2} e^{2} + 120 \, d^{5} m x e - 120 \, d^{6}\right )} b c^{2} e^{\left (m \log \left (x e + d\right ) - 6\right )}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} + \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} x^{7} e^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d x^{6} e^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} x^{5} e^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} x^{4} e^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} x^{3} e^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} x^{2} e^{2} - 720 \, d^{6} m x e + 720 \, d^{7}\right )} c^{3} e^{\left (m \log \left (x e + d\right ) - 7\right )}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

((m^3 + 6*m^2 + 11*m + 6)*x^4*e^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*(m^2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*
d^4)*b^3*e^(m*log(x*e + d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x
^5*e^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*x^3*e^3 + 12*(m^2 + m)*d^3*x^2*e^2
- 24*d^4*m*x*e + 24*d^5)*b^2*c*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) + 3*((m
^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*x^6*e^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*x^5*e^5 - 5*
(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*x^4*e^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*x^3*e^3 - 60*(m^2 + m)*d^4*x^2*e^2 + 120
*d^5*m*x*e - 120*d^6)*b*c^2*e^(m*log(x*e + d) - 6)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720
) + ((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*x^7*e^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 +
274*m^2 + 120*m)*d*x^6*e^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*x^5*e^5 + 30*(m^4 + 6*m^3 + 11*m^2
+ 6*m)*d^3*x^4*e^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*x^3*e^3 + 360*(m^2 + m)*d^5*x^2*e^2 - 720*d^6*m*x*e + 720*d^
7)*c^3*e^(m*log(x*e + d) - 7)/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1253 vs. \(2 (273) = 546\).
time = 1.71, size = 1253, normalized size = 4.69 \begin {gather*} \frac {{\left (720 \, c^{3} d^{7} + {\left ({\left (c^{3} m^{6} + 21 \, c^{3} m^{5} + 175 \, c^{3} m^{4} + 735 \, c^{3} m^{3} + 1624 \, c^{3} m^{2} + 1764 \, c^{3} m + 720 \, c^{3}\right )} x^{7} + 3 \, {\left (b c^{2} m^{6} + 22 \, b c^{2} m^{5} + 190 \, b c^{2} m^{4} + 820 \, b c^{2} m^{3} + 1849 \, b c^{2} m^{2} + 2038 \, b c^{2} m + 840 \, b c^{2}\right )} x^{6} + 3 \, {\left (b^{2} c m^{6} + 23 \, b^{2} c m^{5} + 207 \, b^{2} c m^{4} + 925 \, b^{2} c m^{3} + 2144 \, b^{2} c m^{2} + 2412 \, b^{2} c m + 1008 \, b^{2} c\right )} x^{5} + {\left (b^{3} m^{6} + 24 \, b^{3} m^{5} + 226 \, b^{3} m^{4} + 1056 \, b^{3} m^{3} + 2545 \, b^{3} m^{2} + 2952 \, b^{3} m + 1260 \, b^{3}\right )} x^{4}\right )} e^{7} + {\left ({\left (c^{3} d m^{6} + 15 \, c^{3} d m^{5} + 85 \, c^{3} d m^{4} + 225 \, c^{3} d m^{3} + 274 \, c^{3} d m^{2} + 120 \, c^{3} d m\right )} x^{6} + 3 \, {\left (b c^{2} d m^{6} + 17 \, b c^{2} d m^{5} + 105 \, b c^{2} d m^{4} + 295 \, b c^{2} d m^{3} + 374 \, b c^{2} d m^{2} + 168 \, b c^{2} d m\right )} x^{5} + 3 \, {\left (b^{2} c d m^{6} + 19 \, b^{2} c d m^{5} + 131 \, b^{2} c d m^{4} + 401 \, b^{2} c d m^{3} + 540 \, b^{2} c d m^{2} + 252 \, b^{2} c d m\right )} x^{4} + {\left (b^{3} d m^{6} + 21 \, b^{3} d m^{5} + 163 \, b^{3} d m^{4} + 567 \, b^{3} d m^{3} + 844 \, b^{3} d m^{2} + 420 \, b^{3} d m\right )} x^{3}\right )} e^{6} - 3 \, {\left (2 \, {\left (c^{3} d^{2} m^{5} + 10 \, c^{3} d^{2} m^{4} + 35 \, c^{3} d^{2} m^{3} + 50 \, c^{3} d^{2} m^{2} + 24 \, c^{3} d^{2} m\right )} x^{5} + 5 \, {\left (b c^{2} d^{2} m^{5} + 13 \, b c^{2} d^{2} m^{4} + 53 \, b c^{2} d^{2} m^{3} + 83 \, b c^{2} d^{2} m^{2} + 42 \, b c^{2} d^{2} m\right )} x^{4} + 4 \, {\left (b^{2} c d^{2} m^{5} + 16 \, b^{2} c d^{2} m^{4} + 83 \, b^{2} c d^{2} m^{3} + 152 \, b^{2} c d^{2} m^{2} + 84 \, b^{2} c d^{2} m\right )} x^{3} + {\left (b^{3} d^{2} m^{5} + 19 \, b^{3} d^{2} m^{4} + 125 \, b^{3} d^{2} m^{3} + 317 \, b^{3} d^{2} m^{2} + 210 \, b^{3} d^{2} m\right )} x^{2}\right )} e^{5} + 6 \, {\left (5 \, {\left (c^{3} d^{3} m^{4} + 6 \, c^{3} d^{3} m^{3} + 11 \, c^{3} d^{3} m^{2} + 6 \, c^{3} d^{3} m\right )} x^{4} + 10 \, {\left (b c^{2} d^{3} m^{4} + 10 \, b c^{2} d^{3} m^{3} + 23 \, b c^{2} d^{3} m^{2} + 14 \, b c^{2} d^{3} m\right )} x^{3} + 6 \, {\left (b^{2} c d^{3} m^{4} + 14 \, b^{2} c d^{3} m^{3} + 55 \, b^{2} c d^{3} m^{2} + 42 \, b^{2} c d^{3} m\right )} x^{2} + {\left (b^{3} d^{3} m^{4} + 18 \, b^{3} d^{3} m^{3} + 107 \, b^{3} d^{3} m^{2} + 210 \, b^{3} d^{3} m\right )} x\right )} e^{4} - 6 \, {\left (b^{3} d^{4} m^{3} + 18 \, b^{3} d^{4} m^{2} + 107 \, b^{3} d^{4} m + 210 \, b^{3} d^{4} + 20 \, {\left (c^{3} d^{4} m^{3} + 3 \, c^{3} d^{4} m^{2} + 2 \, c^{3} d^{4} m\right )} x^{3} + 30 \, {\left (b c^{2} d^{4} m^{3} + 8 \, b c^{2} d^{4} m^{2} + 7 \, b c^{2} d^{4} m\right )} x^{2} + 12 \, {\left (b^{2} c d^{4} m^{3} + 13 \, b^{2} c d^{4} m^{2} + 42 \, b^{2} c d^{4} m\right )} x\right )} e^{3} + 72 \, {\left (b^{2} c d^{5} m^{2} + 13 \, b^{2} c d^{5} m + 42 \, b^{2} c d^{5} + 5 \, {\left (c^{3} d^{5} m^{2} + c^{3} d^{5} m\right )} x^{2} + 5 \, {\left (b c^{2} d^{5} m^{2} + 7 \, b c^{2} d^{5} m\right )} x\right )} e^{2} - 360 \, {\left (2 \, c^{3} d^{6} m x + b c^{2} d^{6} m + 7 \, b c^{2} d^{6}\right )} e\right )} {\left (x e + d\right )}^{m} e^{\left (-7\right )}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

(720*c^3*d^7 + ((c^3*m^6 + 21*c^3*m^5 + 175*c^3*m^4 + 735*c^3*m^3 + 1624*c^3*m^2 + 1764*c^3*m + 720*c^3)*x^7 +
3*(b*c^2*m^6 + 22*b*c^2*m^5 + 190*b*c^2*m^4 + 820*b*c^2*m^3 + 1849*b*c^2*m^2 + 2038*b*c^2*m + 840*b*c^2)*x^6
+ 3*(b^2*c*m^6 + 23*b^2*c*m^5 + 207*b^2*c*m^4 + 925*b^2*c*m^3 + 2144*b^2*c*m^2 + 2412*b^2*c*m + 1008*b^2*c)*x^
5 + (b^3*m^6 + 24*b^3*m^5 + 226*b^3*m^4 + 1056*b^3*m^3 + 2545*b^3*m^2 + 2952*b^3*m + 1260*b^3)*x^4)*e^7 + ((c^
3*d*m^6 + 15*c^3*d*m^5 + 85*c^3*d*m^4 + 225*c^3*d*m^3 + 274*c^3*d*m^2 + 120*c^3*d*m)*x^6 + 3*(b*c^2*d*m^6 + 17
*b*c^2*d*m^5 + 105*b*c^2*d*m^4 + 295*b*c^2*d*m^3 + 374*b*c^2*d*m^2 + 168*b*c^2*d*m)*x^5 + 3*(b^2*c*d*m^6 + 19*
b^2*c*d*m^5 + 131*b^2*c*d*m^4 + 401*b^2*c*d*m^3 + 540*b^2*c*d*m^2 + 252*b^2*c*d*m)*x^4 + (b^3*d*m^6 + 21*b^3*d
*m^5 + 163*b^3*d*m^4 + 567*b^3*d*m^3 + 844*b^3*d*m^2 + 420*b^3*d*m)*x^3)*e^6 - 3*(2*(c^3*d^2*m^5 + 10*c^3*d^2*
m^4 + 35*c^3*d^2*m^3 + 50*c^3*d^2*m^2 + 24*c^3*d^2*m)*x^5 + 5*(b*c^2*d^2*m^5 + 13*b*c^2*d^2*m^4 + 53*b*c^2*d^2
*m^3 + 83*b*c^2*d^2*m^2 + 42*b*c^2*d^2*m)*x^4 + 4*(b^2*c*d^2*m^5 + 16*b^2*c*d^2*m^4 + 83*b^2*c*d^2*m^3 + 152*b
^2*c*d^2*m^2 + 84*b^2*c*d^2*m)*x^3 + (b^3*d^2*m^5 + 19*b^3*d^2*m^4 + 125*b^3*d^2*m^3 + 317*b^3*d^2*m^2 + 210*b
^3*d^2*m)*x^2)*e^5 + 6*(5*(c^3*d^3*m^4 + 6*c^3*d^3*m^3 + 11*c^3*d^3*m^2 + 6*c^3*d^3*m)*x^4 + 10*(b*c^2*d^3*m^4
+ 10*b*c^2*d^3*m^3 + 23*b*c^2*d^3*m^2 + 14*b*c^2*d^3*m)*x^3 + 6*(b^2*c*d^3*m^4 + 14*b^2*c*d^3*m^3 + 55*b^2*c*
d^3*m^2 + 42*b^2*c*d^3*m)*x^2 + (b^3*d^3*m^4 + 18*b^3*d^3*m^3 + 107*b^3*d^3*m^2 + 210*b^3*d^3*m)*x)*e^4 - 6*(b
^3*d^4*m^3 + 18*b^3*d^4*m^2 + 107*b^3*d^4*m + 210*b^3*d^4 + 20*(c^3*d^4*m^3 + 3*c^3*d^4*m^2 + 2*c^3*d^4*m)*x^3
+ 30*(b*c^2*d^4*m^3 + 8*b*c^2*d^4*m^2 + 7*b*c^2*d^4*m)*x^2 + 12*(b^2*c*d^4*m^3 + 13*b^2*c*d^4*m^2 + 42*b^2*c*
d^4*m)*x)*e^3 + 72*(b^2*c*d^5*m^2 + 13*b^2*c*d^5*m + 42*b^2*c*d^5 + 5*(c^3*d^5*m^2 + c^3*d^5*m)*x^2 + 5*(b*c^2
*d^5*m^2 + 7*b*c^2*d^5*m)*x)*e^2 - 360*(2*c^3*d^6*m*x + b*c^2*d^6*m + 7*b*c^2*d^6)*e)*(x*e + d)^m*e^(-7)/(m^7
+ 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 21005 vs. \(2 (250) = 500\).
time = 4.61, size = 21005, normalized size = 78.67 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(c*x**2+b*x)**3,x)

[Out]

Piecewise((d**m*(b**3*x**4/4 + 3*b**2*c*x**5/5 + b*c**2*x**6/2 + c**3*x**7/7), Eq(e, 0)), (-b**3*d**3*e**3/(60
*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x
**5 + 60*e**13*x**6) - 6*b**3*d**2*e**4*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**
10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 15*b**3*d*e**5*x**2/(60*d**6*e**7 + 360*d*
*5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6
) - 20*b**3*e**6*x**3/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e
**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 6*b**2*c*d**4*e**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*
e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 36*b**2*c*d**3*e*
*3*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d
*e**12*x**5 + 60*e**13*x**6) - 90*b**2*c*d**2*e**4*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 +
1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 120*b**2*c*d*e**5*x**3/(60*d
**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**
5 + 60*e**13*x**6) - 90*b**2*c*e**6*x**4/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**1
0*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 30*b*c**2*d**5*e/(60*d**6*e**7 + 360*d**5*e
**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) -
180*b*c**2*d**4*e**2*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*
e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 450*b*c**2*d**3*e**3*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 9
00*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 600*b*c**
2*d**2*e**4*x**3/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*
x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 450*b*c**2*d*e**5*x**4/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e
**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 180*b*c**2*e**6*x*
*5/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e
**12*x**5 + 60*e**13*x**6) + 60*c**3*d**6*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 +
1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 147*c**3*d**6/(60*d**6*e**7 +
360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**
13*x**6) + 360*c**3*d**5*e*x*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**
10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 822*c**3*d**5*e*x/(60*d**6*e**7 + 360*d**5
*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6)
+ 900*c**3*d**4*e**2*x**2*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*
x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 1875*c**3*d**4*e**2*x**2/(60*d**6*e**7 + 360*
d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x*
*6) + 1200*c**3*d**3*e**3*x**3*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e
**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 2200*c**3*d**3*e**3*x**3/(60*d**6*e**7 +
360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**
13*x**6) + 900*c**3*d**2*e**4*x**4*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d*
*3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 1350*c**3*d**2*e**4*x**4/(60*d**6*e*
*7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60
*e**13*x**6) + 360*c**3*d*e**5*x**5*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d
**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 360*c**3*d*e**5*x**5/(60*d**6*e**7
+ 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e*
*13*x**6) + 60*c**3*e**6*x**6*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e*
*10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6), Eq(m, -7)), (-b**3*d**3*e**3/(20*d**5*e**7
+ 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 5*b**3*d**
2*e**4*x/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e*
*12*x**5) - 10*b**3*d*e**5*x**2/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 1
00*d*e**11*x**4 + 20*e**12*x**5) - 10*b**3*e**6...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2538 vs. \(2 (273) = 546\).
time = 0.99, size = 2538, normalized size = 9.51 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

((x*e + d)^m*c^3*m^6*x^7*e^7 + (x*e + d)^m*c^3*d*m^6*x^6*e^6 + 3*(x*e + d)^m*b*c^2*m^6*x^6*e^7 + 21*(x*e + d)^
m*c^3*m^5*x^7*e^7 + 3*(x*e + d)^m*b*c^2*d*m^6*x^5*e^6 + 15*(x*e + d)^m*c^3*d*m^5*x^6*e^6 - 6*(x*e + d)^m*c^3*d
^2*m^5*x^5*e^5 + 3*(x*e + d)^m*b^2*c*m^6*x^5*e^7 + 66*(x*e + d)^m*b*c^2*m^5*x^6*e^7 + 175*(x*e + d)^m*c^3*m^4*
x^7*e^7 + 3*(x*e + d)^m*b^2*c*d*m^6*x^4*e^6 + 51*(x*e + d)^m*b*c^2*d*m^5*x^5*e^6 + 85*(x*e + d)^m*c^3*d*m^4*x^
6*e^6 - 15*(x*e + d)^m*b*c^2*d^2*m^5*x^4*e^5 - 60*(x*e + d)^m*c^3*d^2*m^4*x^5*e^5 + 30*(x*e + d)^m*c^3*d^3*m^4
*x^4*e^4 + (x*e + d)^m*b^3*m^6*x^4*e^7 + 69*(x*e + d)^m*b^2*c*m^5*x^5*e^7 + 570*(x*e + d)^m*b*c^2*m^4*x^6*e^7
+ 735*(x*e + d)^m*c^3*m^3*x^7*e^7 + (x*e + d)^m*b^3*d*m^6*x^3*e^6 + 57*(x*e + d)^m*b^2*c*d*m^5*x^4*e^6 + 315*(
x*e + d)^m*b*c^2*d*m^4*x^5*e^6 + 225*(x*e + d)^m*c^3*d*m^3*x^6*e^6 - 12*(x*e + d)^m*b^2*c*d^2*m^5*x^3*e^5 - 19
5*(x*e + d)^m*b*c^2*d^2*m^4*x^4*e^5 - 210*(x*e + d)^m*c^3*d^2*m^3*x^5*e^5 + 60*(x*e + d)^m*b*c^2*d^3*m^4*x^3*e
^4 + 180*(x*e + d)^m*c^3*d^3*m^3*x^4*e^4 - 120*(x*e + d)^m*c^3*d^4*m^3*x^3*e^3 + 24*(x*e + d)^m*b^3*m^5*x^4*e^
7 + 621*(x*e + d)^m*b^2*c*m^4*x^5*e^7 + 2460*(x*e + d)^m*b*c^2*m^3*x^6*e^7 + 1624*(x*e + d)^m*c^3*m^2*x^7*e^7
+ 21*(x*e + d)^m*b^3*d*m^5*x^3*e^6 + 393*(x*e + d)^m*b^2*c*d*m^4*x^4*e^6 + 885*(x*e + d)^m*b*c^2*d*m^3*x^5*e^6
+ 274*(x*e + d)^m*c^3*d*m^2*x^6*e^6 - 3*(x*e + d)^m*b^3*d^2*m^5*x^2*e^5 - 192*(x*e + d)^m*b^2*c*d^2*m^4*x^3*e
^5 - 795*(x*e + d)^m*b*c^2*d^2*m^3*x^4*e^5 - 300*(x*e + d)^m*c^3*d^2*m^2*x^5*e^5 + 36*(x*e + d)^m*b^2*c*d^3*m^
4*x^2*e^4 + 600*(x*e + d)^m*b*c^2*d^3*m^3*x^3*e^4 + 330*(x*e + d)^m*c^3*d^3*m^2*x^4*e^4 - 180*(x*e + d)^m*b*c^
2*d^4*m^3*x^2*e^3 - 360*(x*e + d)^m*c^3*d^4*m^2*x^3*e^3 + 360*(x*e + d)^m*c^3*d^5*m^2*x^2*e^2 + 226*(x*e + d)^
m*b^3*m^4*x^4*e^7 + 2775*(x*e + d)^m*b^2*c*m^3*x^5*e^7 + 5547*(x*e + d)^m*b*c^2*m^2*x^6*e^7 + 1764*(x*e + d)^m
*c^3*m*x^7*e^7 + 163*(x*e + d)^m*b^3*d*m^4*x^3*e^6 + 1203*(x*e + d)^m*b^2*c*d*m^3*x^4*e^6 + 1122*(x*e + d)^m*b
*c^2*d*m^2*x^5*e^6 + 120*(x*e + d)^m*c^3*d*m*x^6*e^6 - 57*(x*e + d)^m*b^3*d^2*m^4*x^2*e^5 - 996*(x*e + d)^m*b^
2*c*d^2*m^3*x^3*e^5 - 1245*(x*e + d)^m*b*c^2*d^2*m^2*x^4*e^5 - 144*(x*e + d)^m*c^3*d^2*m*x^5*e^5 + 6*(x*e + d)
^m*b^3*d^3*m^4*x*e^4 + 504*(x*e + d)^m*b^2*c*d^3*m^3*x^2*e^4 + 1380*(x*e + d)^m*b*c^2*d^3*m^2*x^3*e^4 + 180*(x
*e + d)^m*c^3*d^3*m*x^4*e^4 - 72*(x*e + d)^m*b^2*c*d^4*m^3*x*e^3 - 1440*(x*e + d)^m*b*c^2*d^4*m^2*x^2*e^3 - 24
0*(x*e + d)^m*c^3*d^4*m*x^3*e^3 + 360*(x*e + d)^m*b*c^2*d^5*m^2*x*e^2 + 360*(x*e + d)^m*c^3*d^5*m*x^2*e^2 - 72
0*(x*e + d)^m*c^3*d^6*m*x*e + 1056*(x*e + d)^m*b^3*m^3*x^4*e^7 + 6432*(x*e + d)^m*b^2*c*m^2*x^5*e^7 + 6114*(x*
e + d)^m*b*c^2*m*x^6*e^7 + 720*(x*e + d)^m*c^3*x^7*e^7 + 567*(x*e + d)^m*b^3*d*m^3*x^3*e^6 + 1620*(x*e + d)^m*
b^2*c*d*m^2*x^4*e^6 + 504*(x*e + d)^m*b*c^2*d*m*x^5*e^6 - 375*(x*e + d)^m*b^3*d^2*m^3*x^2*e^5 - 1824*(x*e + d)
^m*b^2*c*d^2*m^2*x^3*e^5 - 630*(x*e + d)^m*b*c^2*d^2*m*x^4*e^5 + 108*(x*e + d)^m*b^3*d^3*m^3*x*e^4 + 1980*(x*e
+ d)^m*b^2*c*d^3*m^2*x^2*e^4 + 840*(x*e + d)^m*b*c^2*d^3*m*x^3*e^4 - 6*(x*e + d)^m*b^3*d^4*m^3*e^3 - 936*(x*e
+ d)^m*b^2*c*d^4*m^2*x*e^3 - 1260*(x*e + d)^m*b*c^2*d^4*m*x^2*e^3 + 72*(x*e + d)^m*b^2*c*d^5*m^2*e^2 + 2520*(
x*e + d)^m*b*c^2*d^5*m*x*e^2 - 360*(x*e + d)^m*b*c^2*d^6*m*e + 720*(x*e + d)^m*c^3*d^7 + 2545*(x*e + d)^m*b^3*
m^2*x^4*e^7 + 7236*(x*e + d)^m*b^2*c*m*x^5*e^7 + 2520*(x*e + d)^m*b*c^2*x^6*e^7 + 844*(x*e + d)^m*b^3*d*m^2*x^
3*e^6 + 756*(x*e + d)^m*b^2*c*d*m*x^4*e^6 - 951*(x*e + d)^m*b^3*d^2*m^2*x^2*e^5 - 1008*(x*e + d)^m*b^2*c*d^2*m
*x^3*e^5 + 642*(x*e + d)^m*b^3*d^3*m^2*x*e^4 + 1512*(x*e + d)^m*b^2*c*d^3*m*x^2*e^4 - 108*(x*e + d)^m*b^3*d^4*
m^2*e^3 - 3024*(x*e + d)^m*b^2*c*d^4*m*x*e^3 + 936*(x*e + d)^m*b^2*c*d^5*m*e^2 - 2520*(x*e + d)^m*b*c^2*d^6*e
+ 2952*(x*e + d)^m*b^3*m*x^4*e^7 + 3024*(x*e + d)^m*b^2*c*x^5*e^7 + 420*(x*e + d)^m*b^3*d*m*x^3*e^6 - 630*(x*e
+ d)^m*b^3*d^2*m*x^2*e^5 + 1260*(x*e + d)^m*b^3*d^3*m*x*e^4 - 642*(x*e + d)^m*b^3*d^4*m*e^3 + 3024*(x*e + d)^
m*b^2*c*d^5*e^2 + 1260*(x*e + d)^m*b^3*x^4*e^7 - 1260*(x*e + d)^m*b^3*d^4*e^3)/(m^7*e^7 + 28*m^6*e^7 + 322*m^5
*e^7 + 1960*m^4*e^7 + 6769*m^3*e^7 + 13132*m^2*e^7 + 13068*m*e^7 + 5040*e^7)

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Mupad [B]
time = 1.02, size = 1085, normalized size = 4.06 \begin {gather*} \frac {c^3\,x^7\,{\left (d+e\,x\right )}^m\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}-\frac {6\,d^4\,{\left (d+e\,x\right )}^m\,\left (b^3\,e^3\,m^3+18\,b^3\,e^3\,m^2+107\,b^3\,e^3\,m+210\,b^3\,e^3-12\,b^2\,c\,d\,e^2\,m^2-156\,b^2\,c\,d\,e^2\,m-504\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e\,m+420\,b\,c^2\,d^2\,e-120\,c^3\,d^3\right )}{e^7\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (b^3\,e^3\,m^3+18\,b^3\,e^3\,m^2+107\,b^3\,e^3\,m+210\,b^3\,e^3+3\,b^2\,c\,d\,e^2\,m^3+39\,b^2\,c\,d\,e^2\,m^2+126\,b^2\,c\,d\,e^2\,m-15\,b\,c^2\,d^2\,e\,m^2-105\,b\,c^2\,d^2\,e\,m+30\,c^3\,d^3\,m\right )}{e^3\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {3\,c\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )\,\left (b^2\,e^2\,m^2+13\,b^2\,e^2\,m+42\,b^2\,e^2+b\,c\,d\,e\,m^2+7\,b\,c\,d\,e\,m-2\,c^2\,d^2\,m\right )}{e^2\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {6\,d^3\,m\,x\,{\left (d+e\,x\right )}^m\,\left (b^3\,e^3\,m^3+18\,b^3\,e^3\,m^2+107\,b^3\,e^3\,m+210\,b^3\,e^3-12\,b^2\,c\,d\,e^2\,m^2-156\,b^2\,c\,d\,e^2\,m-504\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e\,m+420\,b\,c^2\,d^2\,e-120\,c^3\,d^3\right )}{e^6\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {c^2\,x^6\,{\left (d+e\,x\right )}^m\,\left (21\,b\,e+3\,b\,e\,m+c\,d\,m\right )\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{e\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {d\,m\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (b^3\,e^3\,m^3+18\,b^3\,e^3\,m^2+107\,b^3\,e^3\,m+210\,b^3\,e^3-12\,b^2\,c\,d\,e^2\,m^2-156\,b^2\,c\,d\,e^2\,m-504\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e\,m+420\,b\,c^2\,d^2\,e-120\,c^3\,d^3\right )}{e^4\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}-\frac {3\,d^2\,m\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (b^3\,e^3\,m^3+18\,b^3\,e^3\,m^2+107\,b^3\,e^3\,m+210\,b^3\,e^3-12\,b^2\,c\,d\,e^2\,m^2-156\,b^2\,c\,d\,e^2\,m-504\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e\,m+420\,b\,c^2\,d^2\,e-120\,c^3\,d^3\right )}{e^5\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3*(d + e*x)^m,x)

[Out]

(c^3*x^7*(d + e*x)^m*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))/(13068*m + 13132*m^2 + 6769
*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040) - (6*d^4*(d + e*x)^m*(210*b^3*e^3 - 120*c^3*d^3 + 107*b^3*e^3
*m + 18*b^3*e^3*m^2 + b^3*e^3*m^3 + 420*b*c^2*d^2*e - 504*b^2*c*d*e^2 + 60*b*c^2*d^2*e*m - 156*b^2*c*d*e^2*m -
12*b^2*c*d*e^2*m^2))/(e^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (x^4
*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(210*b^3*e^3 + 107*b^3*e^3*m + 30*c^3*d^3*m + 18*b^3*e^3*m^2 + b^3*e^3*m
^3 - 105*b*c^2*d^2*e*m + 126*b^2*c*d*e^2*m - 15*b*c^2*d^2*e*m^2 + 39*b^2*c*d*e^2*m^2 + 3*b^2*c*d*e^2*m^3))/(e^
3*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (3*c*x^5*(d + e*x)^m*(50*m +
35*m^2 + 10*m^3 + m^4 + 24)*(42*b^2*e^2 + 13*b^2*e^2*m - 2*c^2*d^2*m + b^2*e^2*m^2 + 7*b*c*d*e*m + b*c*d*e*m^2
))/(e^2*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (6*d^3*m*x*(d + e*x)^m*
(210*b^3*e^3 - 120*c^3*d^3 + 107*b^3*e^3*m + 18*b^3*e^3*m^2 + b^3*e^3*m^3 + 420*b*c^2*d^2*e - 504*b^2*c*d*e^2
+ 60*b*c^2*d^2*e*m - 156*b^2*c*d*e^2*m - 12*b^2*c*d*e^2*m^2))/(e^6*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4
+ 322*m^5 + 28*m^6 + m^7 + 5040)) + (c^2*x^6*(d + e*x)^m*(21*b*e + 3*b*e*m + c*d*m)*(274*m + 225*m^2 + 85*m^3
+ 15*m^4 + m^5 + 120))/(e*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (d*m*
x^3*(d + e*x)^m*(3*m + m^2 + 2)*(210*b^3*e^3 - 120*c^3*d^3 + 107*b^3*e^3*m + 18*b^3*e^3*m^2 + b^3*e^3*m^3 + 42
0*b*c^2*d^2*e - 504*b^2*c*d*e^2 + 60*b*c^2*d^2*e*m - 156*b^2*c*d*e^2*m - 12*b^2*c*d*e^2*m^2))/(e^4*(13068*m +
13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) - (3*d^2*m*x^2*(m + 1)*(d + e*x)^m*(210*b^3*
e^3 - 120*c^3*d^3 + 107*b^3*e^3*m + 18*b^3*e^3*m^2 + b^3*e^3*m^3 + 420*b*c^2*d^2*e - 504*b^2*c*d*e^2 + 60*b*c^
2*d^2*e*m - 156*b^2*c*d*e^2*m - 12*b^2*c*d*e^2*m^2))/(e^5*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5
+ 28*m^6 + m^7 + 5040))

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