3.5.41 \(\int (d+e x)^m (b x+c x^2)^2 \, dx\) [441]

Optimal. Leaf size=159 \[ \frac {d^2 (c d-b e)^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {2 d (c d-b e) (2 c d-b e) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {2 c (2 c d-b e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^5 (5+m)} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 159, 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 {\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}+\frac {d^2 (c d-b e)^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac {2 d (c d-b e) (2 c d-b e) (d+e x)^{m+2}}{e^5 (m+2)}-\frac {2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^5 (m+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(b*x + c*x^2)^2,x]

[Out]

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

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Mathematica [A]
time = 0.11, size = 138, normalized size = 0.87 \begin {gather*} \frac {(d+e x)^{1+m} \left (\frac {d^2 (c d-b e)^2}{1+m}-\frac {2 d (c d-b e) (2 c d-b e) (d+e x)}{2+m}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^2}{3+m}-\frac {2 c (2 c d-b e) (d+e x)^3}{4+m}+\frac {c^2 (d+e x)^4}{5+m}\right )}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(418\) vs. \(2(159)=318\).
time = 0.47, size = 419, normalized size = 2.64

method	result	size
norman	\(\frac {c^{2} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {\left (b^{2} e^{2} m^{2}+2 b c d e \,m^{2}+9 b^{2} e^{2} m +10 b c d e m -4 c^{2} d^{2} m +20 b^{2} e^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {\left (2 b e m +c d m +10 b e \right ) c \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+9 m +20\right )}+\frac {\left (b^{2} e^{2} m^{2}+9 b^{2} e^{2} m -6 b c d e m +20 b^{2} e^{2}-30 b c d e +12 d^{2} c^{2}\right ) d m \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right )}+\frac {2 d^{3} \left (b^{2} e^{2} m^{2}+9 b^{2} e^{2} m -6 b c d e m +20 b^{2} e^{2}-30 b c d e +12 d^{2} c^{2}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}-\frac {2 m \,d^{2} \left (b^{2} e^{2} m^{2}+9 b^{2} e^{2} m -6 b c d e m +20 b^{2} e^{2}-30 b c d e +12 d^{2} c^{2}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\)	\(419\)
gosper	\(\frac {\left (e x +d \right )^{1+m} \left (c^{2} e^{4} m^{4} x^{4}+2 b c \,e^{4} m^{4} x^{3}+10 c^{2} e^{4} m^{3} x^{4}+b^{2} e^{4} m^{4} x^{2}+22 b c \,e^{4} m^{3} x^{3}-4 c^{2} d \,e^{3} m^{3} x^{3}+35 c^{2} e^{4} m^{2} x^{4}+12 b^{2} e^{4} m^{3} x^{2}-6 b c d \,e^{3} m^{3} x^{2}+82 b c \,e^{4} m^{2} x^{3}-24 c^{2} d \,e^{3} m^{2} x^{3}+50 c^{2} e^{4} m \,x^{4}-2 b^{2} d \,e^{3} m^{3} x +49 b^{2} e^{4} m^{2} x^{2}-48 b c d \,e^{3} m^{2} x^{2}+122 b c \,e^{4} m \,x^{3}+12 c^{2} d^{2} e^{2} m^{2} x^{2}-44 c^{2} d \,e^{3} m \,x^{3}+24 c^{2} x^{4} e^{4}-20 b^{2} d \,e^{3} m^{2} x +78 b^{2} e^{4} m \,x^{2}+12 b c \,d^{2} e^{2} m^{2} x -102 b c d \,e^{3} m \,x^{2}+60 b c \,e^{4} x^{3}+36 c^{2} d^{2} e^{2} m \,x^{2}-24 c^{2} d \,e^{3} x^{3}+2 b^{2} d^{2} e^{2} m^{2}-58 b^{2} d \,e^{3} m x +40 b^{2} e^{4} x^{2}+72 b c \,d^{2} e^{2} m x -60 b c d \,e^{3} x^{2}-24 c^{2} d^{3} e m x +24 c^{2} d^{2} e^{2} x^{2}+18 b^{2} d^{2} e^{2} m -40 b^{2} d \,e^{3} x -12 b c \,d^{3} e m +60 b c \,d^{2} e^{2} x -24 c^{2} d^{3} e x +40 b^{2} d^{2} e^{2}-60 b c \,d^{3} e +24 c^{2} d^{4}\right )}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\)	\(547\)
risch	\(\frac {\left (c^{2} e^{5} m^{4} x^{5}+2 b c \,e^{5} m^{4} x^{4}+c^{2} d \,e^{4} m^{4} x^{4}+10 c^{2} e^{5} m^{3} x^{5}+b^{2} e^{5} m^{4} x^{3}+2 b c d \,e^{4} m^{4} x^{3}+22 b c \,e^{5} m^{3} x^{4}+6 c^{2} d \,e^{4} m^{3} x^{4}+35 c^{2} e^{5} m^{2} x^{5}+b^{2} d \,e^{4} m^{4} x^{2}+12 b^{2} e^{5} m^{3} x^{3}+16 b c d \,e^{4} m^{3} x^{3}+82 b c \,e^{5} m^{2} x^{4}-4 c^{2} d^{2} e^{3} m^{3} x^{3}+11 c^{2} d \,e^{4} m^{2} x^{4}+50 c^{2} e^{5} m \,x^{5}+10 b^{2} d \,e^{4} m^{3} x^{2}+49 b^{2} e^{5} m^{2} x^{3}-6 b c \,d^{2} e^{3} m^{3} x^{2}+34 b c d \,e^{4} m^{2} x^{3}+122 b c \,e^{5} m \,x^{4}-12 c^{2} d^{2} e^{3} m^{2} x^{3}+6 c^{2} d \,e^{4} m \,x^{4}+24 c^{2} x^{5} e^{5}-2 b^{2} d^{2} e^{3} m^{3} x +29 b^{2} d \,e^{4} m^{2} x^{2}+78 b^{2} e^{5} m \,x^{3}-36 b c \,d^{2} e^{3} m^{2} x^{2}+20 b c d \,e^{4} m \,x^{3}+60 b c \,e^{5} x^{4}+12 c^{2} d^{3} e^{2} m^{2} x^{2}-8 c^{2} d^{2} e^{3} m \,x^{3}-18 b^{2} d^{2} e^{3} m^{2} x +20 b^{2} d \,e^{4} m \,x^{2}+40 b^{2} e^{5} x^{3}+12 b c \,d^{3} e^{2} m^{2} x -30 b c \,d^{2} e^{3} m \,x^{2}+12 c^{2} d^{3} e^{2} m \,x^{2}+2 b^{2} d^{3} e^{2} m^{2}-40 b^{2} d^{2} e^{3} m x +60 b c \,d^{3} e^{2} m x -24 c^{2} d^{4} e m x +18 b^{2} d^{3} e^{2} m -12 b c \,d^{4} e m +40 b^{2} d^{3} e^{2}-60 b c \,d^{4} e +24 c^{2} d^{5}\right ) \left (e x +d \right )^{m}}{\left (4+m \right ) \left (5+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) e^{5}}\)	\(663\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2/(5+m)*x^5*exp(m*ln(e*x+d))+(b^2*e^2*m^2+2*b*c*d*e*m^2+9*b^2*e^2*m+10*b*c*d*e*m-4*c^2*d^2*m+20*b^2*e^2)/e^2
/(m^3+12*m^2+47*m+60)*x^3*exp(m*ln(e*x+d))+(2*b*e*m+c*d*m+10*b*e)*c/e/(m^2+9*m+20)*x^4*exp(m*ln(e*x+d))+(b^2*e
^2*m^2+9*b^2*e^2*m-6*b*c*d*e*m+20*b^2*e^2-30*b*c*d*e+12*c^2*d^2)*d/e^3*m/(m^4+14*m^3+71*m^2+154*m+120)*x^2*exp
(m*ln(e*x+d))+2*d^3*(b^2*e^2*m^2+9*b^2*e^2*m-6*b*c*d*e*m+20*b^2*e^2-30*b*c*d*e+12*c^2*d^2)/e^5/(m^5+15*m^4+85*
m^3+225*m^2+274*m+120)*exp(m*ln(e*x+d))-2/e^4*m*d^2*(b^2*e^2*m^2+9*b^2*e^2*m-6*b*c*d*e*m+20*b^2*e^2-30*b*c*d*e
+12*c^2*d^2)/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)*x*exp(m*ln(e*x+d))

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Maxima [A]
time = 0.32, size = 318, normalized size = 2.00 \begin {gather*} \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} x^{3} e^{3} + {\left (m^{2} + m\right )} d x^{2} e^{2} - 2 \, d^{2} m x e + 2 \, d^{3}\right )} b^{2} e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {2 \, {\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 c e^{\left (m \log \left (x e + d\right ) - 4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\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 )} c^{2} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((m^2 + 3*m + 2)*x^3*e^3 + (m^2 + m)*d*x^2*e^2 - 2*d^2*m*x*e + 2*d^3)*b^2*e^(m*log(x*e + d) - 3)/(m^3 + 6*m^2
+ 11*m + 6) + 2*((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*c*e^(m*log(x*e + d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + ((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)*c^2*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m +
120)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (163) = 326\).
time = 1.55, size = 490, normalized size = 3.08 \begin {gather*} \frac {{\left (24 \, c^{2} d^{5} + {\left ({\left (c^{2} m^{4} + 10 \, c^{2} m^{3} + 35 \, c^{2} m^{2} + 50 \, c^{2} m + 24 \, c^{2}\right )} x^{5} + 2 \, {\left (b c m^{4} + 11 \, b c m^{3} + 41 \, b c m^{2} + 61 \, b c m + 30 \, b c\right )} x^{4} + {\left (b^{2} m^{4} + 12 \, b^{2} m^{3} + 49 \, b^{2} m^{2} + 78 \, b^{2} m + 40 \, b^{2}\right )} x^{3}\right )} e^{5} + {\left ({\left (c^{2} d m^{4} + 6 \, c^{2} d m^{3} + 11 \, c^{2} d m^{2} + 6 \, c^{2} d m\right )} x^{4} + 2 \, {\left (b c d m^{4} + 8 \, b c d m^{3} + 17 \, b c d m^{2} + 10 \, b c d m\right )} x^{3} + {\left (b^{2} d m^{4} + 10 \, b^{2} d m^{3} + 29 \, b^{2} d m^{2} + 20 \, b^{2} d m\right )} x^{2}\right )} e^{4} - 2 \, {\left (2 \, {\left (c^{2} d^{2} m^{3} + 3 \, c^{2} d^{2} m^{2} + 2 \, c^{2} d^{2} m\right )} x^{3} + 3 \, {\left (b c d^{2} m^{3} + 6 \, b c d^{2} m^{2} + 5 \, b c d^{2} m\right )} x^{2} + {\left (b^{2} d^{2} m^{3} + 9 \, b^{2} d^{2} m^{2} + 20 \, b^{2} d^{2} m\right )} x\right )} e^{3} + 2 \, {\left (b^{2} d^{3} m^{2} + 9 \, b^{2} d^{3} m + 20 \, b^{2} d^{3} + 6 \, {\left (c^{2} d^{3} m^{2} + c^{2} d^{3} m\right )} x^{2} + 6 \, {\left (b c d^{3} m^{2} + 5 \, b c d^{3} m\right )} x\right )} e^{2} - 12 \, {\left (2 \, c^{2} d^{4} m x + b c d^{4} m + 5 \, b c d^{4}\right )} e\right )} {\left (x e + d\right )}^{m} e^{\left (-5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(24*c^2*d^5 + ((c^2*m^4 + 10*c^2*m^3 + 35*c^2*m^2 + 50*c^2*m + 24*c^2)*x^5 + 2*(b*c*m^4 + 11*b*c*m^3 + 41*b*c*
m^2 + 61*b*c*m + 30*b*c)*x^4 + (b^2*m^4 + 12*b^2*m^3 + 49*b^2*m^2 + 78*b^2*m + 40*b^2)*x^3)*e^5 + ((c^2*d*m^4
+ 6*c^2*d*m^3 + 11*c^2*d*m^2 + 6*c^2*d*m)*x^4 + 2*(b*c*d*m^4 + 8*b*c*d*m^3 + 17*b*c*d*m^2 + 10*b*c*d*m)*x^3 +
(b^2*d*m^4 + 10*b^2*d*m^3 + 29*b^2*d*m^2 + 20*b^2*d*m)*x^2)*e^4 - 2*(2*(c^2*d^2*m^3 + 3*c^2*d^2*m^2 + 2*c^2*d^
2*m)*x^3 + 3*(b*c*d^2*m^3 + 6*b*c*d^2*m^2 + 5*b*c*d^2*m)*x^2 + (b^2*d^2*m^3 + 9*b^2*d^2*m^2 + 20*b^2*d^2*m)*x)
*e^3 + 2*(b^2*d^3*m^2 + 9*b^2*d^3*m + 20*b^2*d^3 + 6*(c^2*d^3*m^2 + c^2*d^3*m)*x^2 + 6*(b*c*d^3*m^2 + 5*b*c*d^
3*m)*x)*e^2 - 12*(2*c^2*d^4*m*x + b*c*d^4*m + 5*b*c*d^4)*e)*(x*e + d)^m*e^(-5)/(m^5 + 15*m^4 + 85*m^3 + 225*m^
2 + 274*m + 120)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 6418 vs. \(2 (144) = 288\).
time = 1.51, size = 6418, normalized size = 40.36 \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)**2,x)

[Out]

Piecewise((d**m*(b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5), Eq(e, 0)), (-b**2*d**2*e**2/(12*d**4*e**5 + 48*d**3*
e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 4*b**2*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x +
72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 6*b**2*e**4*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**
2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 6*b*c*d**3*e/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2
+ 48*d*e**8*x**3 + 12*e**9*x**4) - 24*b*c*d**2*e**2*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48
*d*e**8*x**3 + 12*e**9*x**4) - 36*b*c*d*e**3*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e*
*8*x**3 + 12*e**9*x**4) - 24*b*c*e**4*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3
+ 12*e**9*x**4) + 12*c**2*d**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x*
*3 + 12*e**9*x**4) + 25*c**2*d**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**
9*x**4) + 48*c**2*d**3*e*x*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 +
12*e**9*x**4) + 88*c**2*d**3*e*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9
*x**4) + 72*c**2*d**2*e**2*x**2*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x*
*3 + 12*e**9*x**4) + 108*c**2*d**2*e**2*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x*
*3 + 12*e**9*x**4) + 48*c**2*d*e**3*x**3*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*
d*e**8*x**3 + 12*e**9*x**4) + 48*c**2*d*e**3*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e*
*8*x**3 + 12*e**9*x**4) + 12*c**2*e**4*x**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 +
48*d*e**8*x**3 + 12*e**9*x**4), Eq(m, -5)), (-b**2*d**2*e**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*
e**8*x**3) - 3*b**2*d*e**3*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 3*b**2*e**4*x**2/(3
*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 6*b*c*d**3*e*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e*
*6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 11*b*c*d**3*e/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3
) + 18*b*c*d**2*e**2*x*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 27*b*c*d**2*
e**2*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 18*b*c*d*e**3*x**2*log(d/e + x)/(3*d**3*e
**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 18*b*c*d*e**3*x**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**
7*x**2 + 3*e**8*x**3) + 6*b*c*e**4*x**3*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**
3) - 12*c**2*d**4*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 22*c**2*d**4/(3*d
**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 36*c**2*d**3*e*x*log(d/e + x)/(3*d**3*e**5 + 9*d**2*
e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 54*c**2*d**3*e*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8
*x**3) - 36*c**2*d**2*e**2*x**2*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 36*
c**2*d**2*e**2*x**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 12*c**2*d*e**3*x**3*log(d/e
+ x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 3*c**2*e**4*x**4/(3*d**3*e**5 + 9*d**2*e**6
*x + 9*d*e**7*x**2 + 3*e**8*x**3), Eq(m, -4)), (2*b**2*d**2*e**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e*
*7*x**2) + 3*b**2*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b**2*d*e**3*x*log(d/e + x)/(2*d**2*e*
*5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b**2*d*e**3*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*b**2*e**4*x**2*l
og(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*b*c*d**3*e*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x +
2*e**7*x**2) - 18*b*c*d**3*e/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*b*c*d**2*e**2*x*log(d/e + x)/(2*d**
2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*b*c*d**2*e**2*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*b*c*d*e*
*3*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b*c*e**4*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*
e**7*x**2) + 12*c**2*d**4*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 18*c**2*d**4/(2*d**2*e**5 +
4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d**3*e*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d
**3*e*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*c**2*d**2*e**2*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6
*x + 2*e**7*x**2) - 4*c**2*d*e**3*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + c**2*e**4*x**4/(2*d**2*e**5
+ 4*d*e**6*x + 2*e**7*x**2), Eq(m, -3)), (-6*b**2*d**2*e**2*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 6*b**2*d**2*e
**2/(3*d*e**5 + 3*e**6*x) - 6*b**2*d*e**3*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 3*b**2*e**4*x**2/(3*d*e**5 +
3*e**6*x) + 18*b*c*d**3*e*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 18*b*c*d**3*e/(3*d*e**5 + 3*e**6*x) + 18*b*c*d*
*2*e**2*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 9*b*c*d*e**3*x**2/(3*d*e**5 + 3*e**6*x) + 3*b*c*e**4*x**3/(3*d*
e**5 + 3*e**6*x) - 12*c**2*d**4*log(d/e + x)/(3...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1002 vs. \(2 (163) = 326\).
time = 1.23, size = 1002, normalized size = 6.30 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c^{2} m^{4} x^{5} e^{5} + {\left (x e + d\right )}^{m} c^{2} d m^{4} x^{4} e^{4} + 2 \, {\left (x e + d\right )}^{m} b c m^{4} x^{4} e^{5} + 10 \, {\left (x e + d\right )}^{m} c^{2} m^{3} x^{5} e^{5} + 2 \, {\left (x e + d\right )}^{m} b c d m^{4} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} c^{2} d m^{3} x^{4} e^{4} - 4 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{3} x^{3} e^{3} + {\left (x e + d\right )}^{m} b^{2} m^{4} x^{3} e^{5} + 22 \, {\left (x e + d\right )}^{m} b c m^{3} x^{4} e^{5} + 35 \, {\left (x e + d\right )}^{m} c^{2} m^{2} x^{5} e^{5} + {\left (x e + d\right )}^{m} b^{2} d m^{4} x^{2} e^{4} + 16 \, {\left (x e + d\right )}^{m} b c d m^{3} x^{3} e^{4} + 11 \, {\left (x e + d\right )}^{m} c^{2} d m^{2} x^{4} e^{4} - 6 \, {\left (x e + d\right )}^{m} b c d^{2} m^{3} x^{2} e^{3} - 12 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{2} x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m^{2} x^{2} e^{2} + 12 \, {\left (x e + d\right )}^{m} b^{2} m^{3} x^{3} e^{5} + 82 \, {\left (x e + d\right )}^{m} b c m^{2} x^{4} e^{5} + 50 \, {\left (x e + d\right )}^{m} c^{2} m x^{5} e^{5} + 10 \, {\left (x e + d\right )}^{m} b^{2} d m^{3} x^{2} e^{4} + 34 \, {\left (x e + d\right )}^{m} b c d m^{2} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} c^{2} d m x^{4} e^{4} - 2 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m^{3} x e^{3} - 36 \, {\left (x e + d\right )}^{m} b c d^{2} m^{2} x^{2} e^{3} - 8 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} b c d^{3} m^{2} x e^{2} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m x^{2} e^{2} - 24 \, {\left (x e + d\right )}^{m} c^{2} d^{4} m x e + 49 \, {\left (x e + d\right )}^{m} b^{2} m^{2} x^{3} e^{5} + 122 \, {\left (x e + d\right )}^{m} b c m x^{4} e^{5} + 24 \, {\left (x e + d\right )}^{m} c^{2} x^{5} e^{5} + 29 \, {\left (x e + d\right )}^{m} b^{2} d m^{2} x^{2} e^{4} + 20 \, {\left (x e + d\right )}^{m} b c d m x^{3} e^{4} - 18 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m^{2} x e^{3} - 30 \, {\left (x e + d\right )}^{m} b c d^{2} m x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} b^{2} d^{3} m^{2} e^{2} + 60 \, {\left (x e + d\right )}^{m} b c d^{3} m x e^{2} - 12 \, {\left (x e + d\right )}^{m} b c d^{4} m e + 24 \, {\left (x e + d\right )}^{m} c^{2} d^{5} + 78 \, {\left (x e + d\right )}^{m} b^{2} m x^{3} e^{5} + 60 \, {\left (x e + d\right )}^{m} b c x^{4} e^{5} + 20 \, {\left (x e + d\right )}^{m} b^{2} d m x^{2} e^{4} - 40 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m x e^{3} + 18 \, {\left (x e + d\right )}^{m} b^{2} d^{3} m e^{2} - 60 \, {\left (x e + d\right )}^{m} b c d^{4} e + 40 \, {\left (x e + d\right )}^{m} b^{2} x^{3} e^{5} + 40 \, {\left (x e + d\right )}^{m} b^{2} d^{3} e^{2}}{m^{5} e^{5} + 15 \, m^{4} e^{5} + 85 \, m^{3} e^{5} + 225 \, m^{2} e^{5} + 274 \, m e^{5} + 120 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*c^2*m^4*x^5*e^5 + (x*e + d)^m*c^2*d*m^4*x^4*e^4 + 2*(x*e + d)^m*b*c*m^4*x^4*e^5 + 10*(x*e + d)^m*
c^2*m^3*x^5*e^5 + 2*(x*e + d)^m*b*c*d*m^4*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m^3*x^4*e^4 - 4*(x*e + d)^m*c^2*d^2*m^
3*x^3*e^3 + (x*e + d)^m*b^2*m^4*x^3*e^5 + 22*(x*e + d)^m*b*c*m^3*x^4*e^5 + 35*(x*e + d)^m*c^2*m^2*x^5*e^5 + (x
*e + d)^m*b^2*d*m^4*x^2*e^4 + 16*(x*e + d)^m*b*c*d*m^3*x^3*e^4 + 11*(x*e + d)^m*c^2*d*m^2*x^4*e^4 - 6*(x*e + d
)^m*b*c*d^2*m^3*x^2*e^3 - 12*(x*e + d)^m*c^2*d^2*m^2*x^3*e^3 + 12*(x*e + d)^m*c^2*d^3*m^2*x^2*e^2 + 12*(x*e +
d)^m*b^2*m^3*x^3*e^5 + 82*(x*e + d)^m*b*c*m^2*x^4*e^5 + 50*(x*e + d)^m*c^2*m*x^5*e^5 + 10*(x*e + d)^m*b^2*d*m^
3*x^2*e^4 + 34*(x*e + d)^m*b*c*d*m^2*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m*x^4*e^4 - 2*(x*e + d)^m*b^2*d^2*m^3*x*e^3
- 36*(x*e + d)^m*b*c*d^2*m^2*x^2*e^3 - 8*(x*e + d)^m*c^2*d^2*m*x^3*e^3 + 12*(x*e + d)^m*b*c*d^3*m^2*x*e^2 + 1
2*(x*e + d)^m*c^2*d^3*m*x^2*e^2 - 24*(x*e + d)^m*c^2*d^4*m*x*e + 49*(x*e + d)^m*b^2*m^2*x^3*e^5 + 122*(x*e + d
)^m*b*c*m*x^4*e^5 + 24*(x*e + d)^m*c^2*x^5*e^5 + 29*(x*e + d)^m*b^2*d*m^2*x^2*e^4 + 20*(x*e + d)^m*b*c*d*m*x^3
*e^4 - 18*(x*e + d)^m*b^2*d^2*m^2*x*e^3 - 30*(x*e + d)^m*b*c*d^2*m*x^2*e^3 + 2*(x*e + d)^m*b^2*d^3*m^2*e^2 + 6
0*(x*e + d)^m*b*c*d^3*m*x*e^2 - 12*(x*e + d)^m*b*c*d^4*m*e + 24*(x*e + d)^m*c^2*d^5 + 78*(x*e + d)^m*b^2*m*x^3
*e^5 + 60*(x*e + d)^m*b*c*x^4*e^5 + 20*(x*e + d)^m*b^2*d*m*x^2*e^4 - 40*(x*e + d)^m*b^2*d^2*m*x*e^3 + 18*(x*e
+ d)^m*b^2*d^3*m*e^2 - 60*(x*e + d)^m*b*c*d^4*e + 40*(x*e + d)^m*b^2*x^3*e^5 + 40*(x*e + d)^m*b^2*d^3*e^2)/(m^
5*e^5 + 15*m^4*e^5 + 85*m^3*e^5 + 225*m^2*e^5 + 274*m*e^5 + 120*e^5)

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Mupad [B]
time = 0.61, size = 464, normalized size = 2.92 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {c^2\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {2\,d^3\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2-6\,b\,c\,d\,e\,m-30\,b\,c\,d\,e+12\,c^2\,d^2\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x^3\,\left (m^2+3\,m+2\right )\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2+2\,b\,c\,d\,e\,m^2+10\,b\,c\,d\,e\,m-4\,c^2\,d^2\,m\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c\,x^4\,\left (10\,b\,e+2\,b\,e\,m+c\,d\,m\right )\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}-\frac {2\,d^2\,m\,x\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2-6\,b\,c\,d\,e\,m-30\,b\,c\,d\,e+12\,c^2\,d^2\right )}{e^4\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {d\,m\,x^2\,\left (m+1\right )\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2-6\,b\,c\,d\,e\,m-30\,b\,c\,d\,e+12\,c^2\,d^2\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2*(d + e*x)^m,x)

[Out]

(d + e*x)^m*((c^2*x^5*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (
2*d^3*(20*b^2*e^2 + 12*c^2*d^2 + 9*b^2*e^2*m + b^2*e^2*m^2 - 30*b*c*d*e - 6*b*c*d*e*m))/(e^5*(274*m + 225*m^2
+ 85*m^3 + 15*m^4 + m^5 + 120)) + (x^3*(3*m + m^2 + 2)*(20*b^2*e^2 + 9*b^2*e^2*m - 4*c^2*d^2*m + b^2*e^2*m^2 +
10*b*c*d*e*m + 2*b*c*d*e*m^2))/(e^2*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (c*x^4*(10*b*e + 2*b*e
*m + c*d*m)*(11*m + 6*m^2 + m^3 + 6))/(e*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) - (2*d^2*m*x*(20*b^2
*e^2 + 12*c^2*d^2 + 9*b^2*e^2*m + b^2*e^2*m^2 - 30*b*c*d*e - 6*b*c*d*e*m))/(e^4*(274*m + 225*m^2 + 85*m^3 + 15
*m^4 + m^5 + 120)) + (d*m*x^2*(m + 1)*(20*b^2*e^2 + 12*c^2*d^2 + 9*b^2*e^2*m + b^2*e^2*m^2 - 30*b*c*d*e - 6*b*
c*d*e*m))/(e^3*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)))

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