3.18.1 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=187 \[ -\frac {7 b^6 (d+e x)^3 (b d-a e)}{3 e^8}+\frac {21 b^5 (d+e x)^2 (b d-a e)^2}{2 e^8}-\frac {35 b^4 x (b d-a e)^3}{e^7}+\frac {35 b^3 (b d-a e)^4 \log (d+e x)}{e^8}+\frac {21 b^2 (b d-a e)^5}{e^8 (d+e x)}-\frac {7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac {(b d-a e)^7}{3 e^8 (d+e x)^3}+\frac {b^7 (d+e x)^4}{4 e^8} \]

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Rubi [A]  time = 0.21, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \begin {gather*} -\frac {7 b^6 (d+e x)^3 (b d-a e)}{3 e^8}+\frac {21 b^5 (d+e x)^2 (b d-a e)^2}{2 e^8}-\frac {35 b^4 x (b d-a e)^3}{e^7}+\frac {21 b^2 (b d-a e)^5}{e^8 (d+e x)}+\frac {35 b^3 (b d-a e)^4 \log (d+e x)}{e^8}-\frac {7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac {(b d-a e)^7}{3 e^8 (d+e x)^3}+\frac {b^7 (d+e x)^4}{4 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^4,x]

[Out]

(-35*b^4*(b*d - a*e)^3*x)/e^7 + (b*d - a*e)^7/(3*e^8*(d + e*x)^3) - (7*b*(b*d - a*e)^6)/(2*e^8*(d + e*x)^2) +
(21*b^2*(b*d - a*e)^5)/(e^8*(d + e*x)) + (21*b^5*(b*d - a*e)^2*(d + e*x)^2)/(2*e^8) - (7*b^6*(b*d - a*e)*(d +
e*x)^3)/(3*e^8) + (b^7*(d + e*x)^4)/(4*e^8) + (35*b^3*(b*d - a*e)^4*Log[d + e*x])/e^8

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx &=\int \frac {(a+b x)^7}{(d+e x)^4} \, dx\\ &=\int \left (-\frac {35 b^4 (b d-a e)^3}{e^7}+\frac {(-b d+a e)^7}{e^7 (d+e x)^4}+\frac {7 b (b d-a e)^6}{e^7 (d+e x)^3}-\frac {21 b^2 (b d-a e)^5}{e^7 (d+e x)^2}+\frac {35 b^3 (b d-a e)^4}{e^7 (d+e x)}+\frac {21 b^5 (b d-a e)^2 (d+e x)}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^2}{e^7}+\frac {b^7 (d+e x)^3}{e^7}\right ) \, dx\\ &=-\frac {35 b^4 (b d-a e)^3 x}{e^7}+\frac {(b d-a e)^7}{3 e^8 (d+e x)^3}-\frac {7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac {21 b^2 (b d-a e)^5}{e^8 (d+e x)}+\frac {21 b^5 (b d-a e)^2 (d+e x)^2}{2 e^8}-\frac {7 b^6 (b d-a e) (d+e x)^3}{3 e^8}+\frac {b^7 (d+e x)^4}{4 e^8}+\frac {35 b^3 (b d-a e)^4 \log (d+e x)}{e^8}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 199, normalized size = 1.06 \begin {gather*} \frac {6 b^5 e^2 x^2 \left (21 a^2 e^2-28 a b d e+10 b^2 d^2\right )-12 b^4 e x \left (-35 a^3 e^3+84 a^2 b d e^2-70 a b^2 d^2 e+20 b^3 d^3\right )-4 b^6 e^3 x^3 (4 b d-7 a e)+420 b^3 (b d-a e)^4 \log (d+e x)+\frac {252 b^2 (b d-a e)^5}{d+e x}-\frac {42 b (b d-a e)^6}{(d+e x)^2}+\frac {4 (b d-a e)^7}{(d+e x)^3}+3 b^7 e^4 x^4}{12 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^4,x]

[Out]

(-12*b^4*e*(20*b^3*d^3 - 70*a*b^2*d^2*e + 84*a^2*b*d*e^2 - 35*a^3*e^3)*x + 6*b^5*e^2*(10*b^2*d^2 - 28*a*b*d*e
+ 21*a^2*e^2)*x^2 - 4*b^6*e^3*(4*b*d - 7*a*e)*x^3 + 3*b^7*e^4*x^4 + (4*(b*d - a*e)^7)/(d + e*x)^3 - (42*b*(b*d
- a*e)^6)/(d + e*x)^2 + (252*b^2*(b*d - a*e)^5)/(d + e*x) + 420*b^3*(b*d - a*e)^4*Log[d + e*x])/(12*e^8)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^4,x]

[Out]

IntegrateAlgebraic[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^4, x]

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fricas [B]  time = 0.43, size = 737, normalized size = 3.94 \begin {gather*} \frac {3 \, b^{7} e^{7} x^{7} + 214 \, b^{7} d^{7} - 1036 \, a b^{6} d^{6} e + 1974 \, a^{2} b^{5} d^{5} e^{2} - 1820 \, a^{3} b^{4} d^{4} e^{3} + 770 \, a^{4} b^{3} d^{3} e^{4} - 84 \, a^{5} b^{2} d^{2} e^{5} - 14 \, a^{6} b d e^{6} - 4 \, a^{7} e^{7} - 7 \, {\left (b^{7} d e^{6} - 4 \, a b^{6} e^{7}\right )} x^{6} + 21 \, {\left (b^{7} d^{2} e^{5} - 4 \, a b^{6} d e^{6} + 6 \, a^{2} b^{5} e^{7}\right )} x^{5} - 105 \, {\left (b^{7} d^{3} e^{4} - 4 \, a b^{6} d^{2} e^{5} + 6 \, a^{2} b^{5} d e^{6} - 4 \, a^{3} b^{4} e^{7}\right )} x^{4} - 2 \, {\left (278 \, b^{7} d^{4} e^{3} - 1022 \, a b^{6} d^{3} e^{4} + 1323 \, a^{2} b^{5} d^{2} e^{5} - 630 \, a^{3} b^{4} d e^{6}\right )} x^{3} - 6 \, {\left (68 \, b^{7} d^{5} e^{2} - 182 \, a b^{6} d^{4} e^{3} + 63 \, a^{2} b^{5} d^{3} e^{4} + 210 \, a^{3} b^{4} d^{2} e^{5} - 210 \, a^{4} b^{3} d e^{6} + 42 \, a^{5} b^{2} e^{7}\right )} x^{2} + 6 \, {\left (37 \, b^{7} d^{6} e - 238 \, a b^{6} d^{5} e^{2} + 567 \, a^{2} b^{5} d^{4} e^{3} - 630 \, a^{3} b^{4} d^{3} e^{4} + 315 \, a^{4} b^{3} d^{2} e^{5} - 42 \, a^{5} b^{2} d e^{6} - 7 \, a^{6} b e^{7}\right )} x + 420 \, {\left (b^{7} d^{7} - 4 \, a b^{6} d^{6} e + 6 \, a^{2} b^{5} d^{5} e^{2} - 4 \, a^{3} b^{4} d^{4} e^{3} + a^{4} b^{3} d^{3} e^{4} + {\left (b^{7} d^{4} e^{3} - 4 \, a b^{6} d^{3} e^{4} + 6 \, a^{2} b^{5} d^{2} e^{5} - 4 \, a^{3} b^{4} d e^{6} + a^{4} b^{3} e^{7}\right )} x^{3} + 3 \, {\left (b^{7} d^{5} e^{2} - 4 \, a b^{6} d^{4} e^{3} + 6 \, a^{2} b^{5} d^{3} e^{4} - 4 \, a^{3} b^{4} d^{2} e^{5} + a^{4} b^{3} d e^{6}\right )} x^{2} + 3 \, {\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 6 \, a^{2} b^{5} d^{4} e^{3} - 4 \, a^{3} b^{4} d^{3} e^{4} + a^{4} b^{3} d^{2} e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/12*(3*b^7*e^7*x^7 + 214*b^7*d^7 - 1036*a*b^6*d^6*e + 1974*a^2*b^5*d^5*e^2 - 1820*a^3*b^4*d^4*e^3 + 770*a^4*b
^3*d^3*e^4 - 84*a^5*b^2*d^2*e^5 - 14*a^6*b*d*e^6 - 4*a^7*e^7 - 7*(b^7*d*e^6 - 4*a*b^6*e^7)*x^6 + 21*(b^7*d^2*e
^5 - 4*a*b^6*d*e^6 + 6*a^2*b^5*e^7)*x^5 - 105*(b^7*d^3*e^4 - 4*a*b^6*d^2*e^5 + 6*a^2*b^5*d*e^6 - 4*a^3*b^4*e^7
)*x^4 - 2*(278*b^7*d^4*e^3 - 1022*a*b^6*d^3*e^4 + 1323*a^2*b^5*d^2*e^5 - 630*a^3*b^4*d*e^6)*x^3 - 6*(68*b^7*d^
5*e^2 - 182*a*b^6*d^4*e^3 + 63*a^2*b^5*d^3*e^4 + 210*a^3*b^4*d^2*e^5 - 210*a^4*b^3*d*e^6 + 42*a^5*b^2*e^7)*x^2
+ 6*(37*b^7*d^6*e - 238*a*b^6*d^5*e^2 + 567*a^2*b^5*d^4*e^3 - 630*a^3*b^4*d^3*e^4 + 315*a^4*b^3*d^2*e^5 - 42*
a^5*b^2*d*e^6 - 7*a^6*b*e^7)*x + 420*(b^7*d^7 - 4*a*b^6*d^6*e + 6*a^2*b^5*d^5*e^2 - 4*a^3*b^4*d^4*e^3 + a^4*b^
3*d^3*e^4 + (b^7*d^4*e^3 - 4*a*b^6*d^3*e^4 + 6*a^2*b^5*d^2*e^5 - 4*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x^3 + 3*(b^7*d
^5*e^2 - 4*a*b^6*d^4*e^3 + 6*a^2*b^5*d^3*e^4 - 4*a^3*b^4*d^2*e^5 + a^4*b^3*d*e^6)*x^2 + 3*(b^7*d^6*e - 4*a*b^6
*d^5*e^2 + 6*a^2*b^5*d^4*e^3 - 4*a^3*b^4*d^3*e^4 + a^4*b^3*d^2*e^5)*x)*log(e*x + d))/(e^11*x^3 + 3*d*e^10*x^2
+ 3*d^2*e^9*x + d^3*e^8)

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giac [B]  time = 0.18, size = 442, normalized size = 2.36 \begin {gather*} 35 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, b^{7} x^{4} e^{12} - 16 \, b^{7} d x^{3} e^{11} + 60 \, b^{7} d^{2} x^{2} e^{10} - 240 \, b^{7} d^{3} x e^{9} + 28 \, a b^{6} x^{3} e^{12} - 168 \, a b^{6} d x^{2} e^{11} + 840 \, a b^{6} d^{2} x e^{10} + 126 \, a^{2} b^{5} x^{2} e^{12} - 1008 \, a^{2} b^{5} d x e^{11} + 420 \, a^{3} b^{4} x e^{12}\right )} e^{\left (-16\right )} + \frac {{\left (107 \, b^{7} d^{7} - 518 \, a b^{6} d^{6} e + 987 \, a^{2} b^{5} d^{5} e^{2} - 910 \, a^{3} b^{4} d^{4} e^{3} + 385 \, a^{4} b^{3} d^{3} e^{4} - 42 \, a^{5} b^{2} d^{2} e^{5} - 7 \, a^{6} b d e^{6} - 2 \, a^{7} e^{7} + 126 \, {\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{2} + 21 \, {\left (11 \, b^{7} d^{6} e - 54 \, a b^{6} d^{5} e^{2} + 105 \, a^{2} b^{5} d^{4} e^{3} - 100 \, a^{3} b^{4} d^{3} e^{4} + 45 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x, algorithm="giac")

[Out]

35*(b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*e^(-8)*log(abs(x*e + d)) + 1/
12*(3*b^7*x^4*e^12 - 16*b^7*d*x^3*e^11 + 60*b^7*d^2*x^2*e^10 - 240*b^7*d^3*x*e^9 + 28*a*b^6*x^3*e^12 - 168*a*b
^6*d*x^2*e^11 + 840*a*b^6*d^2*x*e^10 + 126*a^2*b^5*x^2*e^12 - 1008*a^2*b^5*d*x*e^11 + 420*a^3*b^4*x*e^12)*e^(-
16) + 1/6*(107*b^7*d^7 - 518*a*b^6*d^6*e + 987*a^2*b^5*d^5*e^2 - 910*a^3*b^4*d^4*e^3 + 385*a^4*b^3*d^3*e^4 - 4
2*a^5*b^2*d^2*e^5 - 7*a^6*b*d*e^6 - 2*a^7*e^7 + 126*(b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a
^3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^2 + 21*(11*b^7*d^6*e - 54*a*b^6*d^5*e^2 + 105*a^2*b^5*d^4*e^
3 - 100*a^3*b^4*d^3*e^4 + 45*a^4*b^3*d^2*e^5 - 6*a^5*b^2*d*e^6 - a^6*b*e^7)*x)*e^(-8)/(x*e + d)^3

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maple [B]  time = 0.07, size = 622, normalized size = 3.33 \begin {gather*} \frac {b^{7} x^{4}}{4 e^{4}}-\frac {a^{7}}{3 \left (e x +d \right )^{3} e}+\frac {7 a^{6} b d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {7 a^{5} b^{2} d^{2}}{\left (e x +d \right )^{3} e^{3}}+\frac {35 a^{4} b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {35 a^{3} b^{4} d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {7 a^{2} b^{5} d^{5}}{\left (e x +d \right )^{3} e^{6}}-\frac {7 a \,b^{6} d^{6}}{3 \left (e x +d \right )^{3} e^{7}}+\frac {7 a \,b^{6} x^{3}}{3 e^{4}}+\frac {b^{7} d^{7}}{3 \left (e x +d \right )^{3} e^{8}}-\frac {4 b^{7} d \,x^{3}}{3 e^{5}}-\frac {7 a^{6} b}{2 \left (e x +d \right )^{2} e^{2}}+\frac {21 a^{5} b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {105 a^{4} b^{3} d^{2}}{2 \left (e x +d \right )^{2} e^{4}}+\frac {70 a^{3} b^{4} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {105 a^{2} b^{5} d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {21 a^{2} b^{5} x^{2}}{2 e^{4}}+\frac {21 a \,b^{6} d^{5}}{\left (e x +d \right )^{2} e^{7}}-\frac {14 a \,b^{6} d \,x^{2}}{e^{5}}-\frac {7 b^{7} d^{6}}{2 \left (e x +d \right )^{2} e^{8}}+\frac {5 b^{7} d^{2} x^{2}}{e^{6}}-\frac {21 a^{5} b^{2}}{\left (e x +d \right ) e^{3}}+\frac {105 a^{4} b^{3} d}{\left (e x +d \right ) e^{4}}+\frac {35 a^{4} b^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {210 a^{3} b^{4} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {140 a^{3} b^{4} d \ln \left (e x +d \right )}{e^{5}}+\frac {35 a^{3} b^{4} x}{e^{4}}+\frac {210 a^{2} b^{5} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {210 a^{2} b^{5} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {84 a^{2} b^{5} d x}{e^{5}}-\frac {105 a \,b^{6} d^{4}}{\left (e x +d \right ) e^{7}}-\frac {140 a \,b^{6} d^{3} \ln \left (e x +d \right )}{e^{7}}+\frac {70 a \,b^{6} d^{2} x}{e^{6}}+\frac {21 b^{7} d^{5}}{\left (e x +d \right ) e^{8}}+\frac {35 b^{7} d^{4} \ln \left (e x +d \right )}{e^{8}}-\frac {20 b^{7} d^{3} x}{e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x)

[Out]

35*b^4/e^4*a^3*x-20*b^7/e^7*d^3*x-21*b^2/e^3/(e*x+d)*a^5+21*b^7/e^8/(e*x+d)*d^5+1/3/e^8/(e*x+d)^3*b^7*d^7-7/2*
b/e^2/(e*x+d)^2*a^6-7/2*b^7/e^8/(e*x+d)^2*d^6+1/4*b^7/e^4*x^4-1/3/e/(e*x+d)^3*a^7+70*b^4/e^5/(e*x+d)^2*a^3*d^3
+35*b^3/e^4*ln(e*x+d)*a^4+35*b^7/e^8*ln(e*x+d)*d^4+7/3*b^6/e^4*x^3*a-4/3*b^7/e^5*x^3*d+21/2*b^5/e^4*x^2*a^2+5*
b^7/e^6*x^2*d^2+210*b^5/e^6/(e*x+d)*a^2*d^3-105*b^6/e^7/(e*x+d)*a*d^4+21*b^2/e^3/(e*x+d)^2*a^5*d-140*b^4/e^5*l
n(e*x+d)*a^3*d+210*b^5/e^6*ln(e*x+d)*a^2*d^2-140*b^6/e^7*ln(e*x+d)*a*d^3+21*b^6/e^7/(e*x+d)^2*a*d^5+7/3/e^2/(e
*x+d)^3*d*a^6*b-7/e^3/(e*x+d)^3*d^2*a^5*b^2+35/3/e^4/(e*x+d)^3*d^3*a^4*b^3-35/3/e^5/(e*x+d)^3*a^3*b^4*d^4+7/e^
6/(e*x+d)^3*a^2*b^5*d^5-7/3/e^7/(e*x+d)^3*a*b^6*d^6-14*b^6/e^5*x^2*a*d-84*b^5/e^5*a^2*d*x+70*b^6/e^6*a*d^2*x+1
05*b^3/e^4/(e*x+d)*a^4*d-210*b^4/e^5/(e*x+d)*a^3*d^2-105/2*b^3/e^4/(e*x+d)^2*a^4*d^2-105/2*b^5/e^6/(e*x+d)^2*a
^2*d^4

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maxima [B]  time = 0.67, size = 485, normalized size = 2.59 \begin {gather*} \frac {107 \, b^{7} d^{7} - 518 \, a b^{6} d^{6} e + 987 \, a^{2} b^{5} d^{5} e^{2} - 910 \, a^{3} b^{4} d^{4} e^{3} + 385 \, a^{4} b^{3} d^{3} e^{4} - 42 \, a^{5} b^{2} d^{2} e^{5} - 7 \, a^{6} b d e^{6} - 2 \, a^{7} e^{7} + 126 \, {\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{2} + 21 \, {\left (11 \, b^{7} d^{6} e - 54 \, a b^{6} d^{5} e^{2} + 105 \, a^{2} b^{5} d^{4} e^{3} - 100 \, a^{3} b^{4} d^{3} e^{4} + 45 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x}{6 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac {3 \, b^{7} e^{3} x^{4} - 4 \, {\left (4 \, b^{7} d e^{2} - 7 \, a b^{6} e^{3}\right )} x^{3} + 6 \, {\left (10 \, b^{7} d^{2} e - 28 \, a b^{6} d e^{2} + 21 \, a^{2} b^{5} e^{3}\right )} x^{2} - 12 \, {\left (20 \, b^{7} d^{3} - 70 \, a b^{6} d^{2} e + 84 \, a^{2} b^{5} d e^{2} - 35 \, a^{3} b^{4} e^{3}\right )} x}{12 \, e^{7}} + \frac {35 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*(107*b^7*d^7 - 518*a*b^6*d^6*e + 987*a^2*b^5*d^5*e^2 - 910*a^3*b^4*d^4*e^3 + 385*a^4*b^3*d^3*e^4 - 42*a^5*
b^2*d^2*e^5 - 7*a^6*b*d*e^6 - 2*a^7*e^7 + 126*(b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4
*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^2 + 21*(11*b^7*d^6*e - 54*a*b^6*d^5*e^2 + 105*a^2*b^5*d^4*e^3 - 10
0*a^3*b^4*d^3*e^4 + 45*a^4*b^3*d^2*e^5 - 6*a^5*b^2*d*e^6 - a^6*b*e^7)*x)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*
x + d^3*e^8) + 1/12*(3*b^7*e^3*x^4 - 4*(4*b^7*d*e^2 - 7*a*b^6*e^3)*x^3 + 6*(10*b^7*d^2*e - 28*a*b^6*d*e^2 + 21
*a^2*b^5*e^3)*x^2 - 12*(20*b^7*d^3 - 70*a*b^6*d^2*e + 84*a^2*b^5*d*e^2 - 35*a^3*b^4*e^3)*x)/e^7 + 35*(b^7*d^4
- 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*log(e*x + d)/e^8

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mupad [B]  time = 2.12, size = 558, normalized size = 2.98 \begin {gather*} x\,\left (\frac {35\,a^3\,b^4}{e^4}-\frac {4\,b^7\,d^3}{e^7}+\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {7\,a\,b^6}{e^4}-\frac {4\,b^7\,d}{e^5}\right )}{e}-\frac {21\,a^2\,b^5}{e^4}+\frac {6\,b^7\,d^2}{e^6}\right )}{e}-\frac {6\,d^2\,\left (\frac {7\,a\,b^6}{e^4}-\frac {4\,b^7\,d}{e^5}\right )}{e^2}\right )-\frac {\frac {2\,a^7\,e^7+7\,a^6\,b\,d\,e^6+42\,a^5\,b^2\,d^2\,e^5-385\,a^4\,b^3\,d^3\,e^4+910\,a^3\,b^4\,d^4\,e^3-987\,a^2\,b^5\,d^5\,e^2+518\,a\,b^6\,d^6\,e-107\,b^7\,d^7}{6\,e}+x\,\left (\frac {7\,a^6\,b\,e^6}{2}+21\,a^5\,b^2\,d\,e^5-\frac {315\,a^4\,b^3\,d^2\,e^4}{2}+350\,a^3\,b^4\,d^3\,e^3-\frac {735\,a^2\,b^5\,d^4\,e^2}{2}+189\,a\,b^6\,d^5\,e-\frac {77\,b^7\,d^6}{2}\right )-x^2\,\left (-21\,a^5\,b^2\,e^6+105\,a^4\,b^3\,d\,e^5-210\,a^3\,b^4\,d^2\,e^4+210\,a^2\,b^5\,d^3\,e^3-105\,a\,b^6\,d^4\,e^2+21\,b^7\,d^5\,e\right )}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}+x^3\,\left (\frac {7\,a\,b^6}{3\,e^4}-\frac {4\,b^7\,d}{3\,e^5}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^4}-\frac {4\,b^7\,d}{e^5}\right )}{e}-\frac {21\,a^2\,b^5}{2\,e^4}+\frac {3\,b^7\,d^2}{e^6}\right )+\frac {\ln \left (d+e\,x\right )\,\left (35\,a^4\,b^3\,e^4-140\,a^3\,b^4\,d\,e^3+210\,a^2\,b^5\,d^2\,e^2-140\,a\,b^6\,d^3\,e+35\,b^7\,d^4\right )}{e^8}+\frac {b^7\,x^4}{4\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/(d + e*x)^4,x)

[Out]

x*((35*a^3*b^4)/e^4 - (4*b^7*d^3)/e^7 + (4*d*((4*d*((7*a*b^6)/e^4 - (4*b^7*d)/e^5))/e - (21*a^2*b^5)/e^4 + (6*
b^7*d^2)/e^6))/e - (6*d^2*((7*a*b^6)/e^4 - (4*b^7*d)/e^5))/e^2) - ((2*a^7*e^7 - 107*b^7*d^7 - 987*a^2*b^5*d^5*
e^2 + 910*a^3*b^4*d^4*e^3 - 385*a^4*b^3*d^3*e^4 + 42*a^5*b^2*d^2*e^5 + 518*a*b^6*d^6*e + 7*a^6*b*d*e^6)/(6*e)
+ x*((7*a^6*b*e^6)/2 - (77*b^7*d^6)/2 + 21*a^5*b^2*d*e^5 - (735*a^2*b^5*d^4*e^2)/2 + 350*a^3*b^4*d^3*e^3 - (31
5*a^4*b^3*d^2*e^4)/2 + 189*a*b^6*d^5*e) - x^2*(21*b^7*d^5*e - 21*a^5*b^2*e^6 - 105*a*b^6*d^4*e^2 + 105*a^4*b^3
*d*e^5 + 210*a^2*b^5*d^3*e^3 - 210*a^3*b^4*d^2*e^4))/(d^3*e^7 + e^10*x^3 + 3*d^2*e^8*x + 3*d*e^9*x^2) + x^3*((
7*a*b^6)/(3*e^4) - (4*b^7*d)/(3*e^5)) - x^2*((2*d*((7*a*b^6)/e^4 - (4*b^7*d)/e^5))/e - (21*a^2*b^5)/(2*e^4) +
(3*b^7*d^2)/e^6) + (log(d + e*x)*(35*b^7*d^4 + 35*a^4*b^3*e^4 - 140*a^3*b^4*d*e^3 + 210*a^2*b^5*d^2*e^2 - 140*
a*b^6*d^3*e))/e^8 + (b^7*x^4)/(4*e^4)

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sympy [B]  time = 6.27, size = 474, normalized size = 2.53 \begin {gather*} \frac {b^{7} x^{4}}{4 e^{4}} + \frac {35 b^{3} \left (a e - b d\right )^{4} \log {\left (d + e x \right )}}{e^{8}} + x^{3} \left (\frac {7 a b^{6}}{3 e^{4}} - \frac {4 b^{7} d}{3 e^{5}}\right ) + x^{2} \left (\frac {21 a^{2} b^{5}}{2 e^{4}} - \frac {14 a b^{6} d}{e^{5}} + \frac {5 b^{7} d^{2}}{e^{6}}\right ) + x \left (\frac {35 a^{3} b^{4}}{e^{4}} - \frac {84 a^{2} b^{5} d}{e^{5}} + \frac {70 a b^{6} d^{2}}{e^{6}} - \frac {20 b^{7} d^{3}}{e^{7}}\right ) + \frac {- 2 a^{7} e^{7} - 7 a^{6} b d e^{6} - 42 a^{5} b^{2} d^{2} e^{5} + 385 a^{4} b^{3} d^{3} e^{4} - 910 a^{3} b^{4} d^{4} e^{3} + 987 a^{2} b^{5} d^{5} e^{2} - 518 a b^{6} d^{6} e + 107 b^{7} d^{7} + x^{2} \left (- 126 a^{5} b^{2} e^{7} + 630 a^{4} b^{3} d e^{6} - 1260 a^{3} b^{4} d^{2} e^{5} + 1260 a^{2} b^{5} d^{3} e^{4} - 630 a b^{6} d^{4} e^{3} + 126 b^{7} d^{5} e^{2}\right ) + x \left (- 21 a^{6} b e^{7} - 126 a^{5} b^{2} d e^{6} + 945 a^{4} b^{3} d^{2} e^{5} - 2100 a^{3} b^{4} d^{3} e^{4} + 2205 a^{2} b^{5} d^{4} e^{3} - 1134 a b^{6} d^{5} e^{2} + 231 b^{7} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**4,x)

[Out]

b**7*x**4/(4*e**4) + 35*b**3*(a*e - b*d)**4*log(d + e*x)/e**8 + x**3*(7*a*b**6/(3*e**4) - 4*b**7*d/(3*e**5)) +
x**2*(21*a**2*b**5/(2*e**4) - 14*a*b**6*d/e**5 + 5*b**7*d**2/e**6) + x*(35*a**3*b**4/e**4 - 84*a**2*b**5*d/e*
*5 + 70*a*b**6*d**2/e**6 - 20*b**7*d**3/e**7) + (-2*a**7*e**7 - 7*a**6*b*d*e**6 - 42*a**5*b**2*d**2*e**5 + 385
*a**4*b**3*d**3*e**4 - 910*a**3*b**4*d**4*e**3 + 987*a**2*b**5*d**5*e**2 - 518*a*b**6*d**6*e + 107*b**7*d**7 +
x**2*(-126*a**5*b**2*e**7 + 630*a**4*b**3*d*e**6 - 1260*a**3*b**4*d**2*e**5 + 1260*a**2*b**5*d**3*e**4 - 630*
a*b**6*d**4*e**3 + 126*b**7*d**5*e**2) + x*(-21*a**6*b*e**7 - 126*a**5*b**2*d*e**6 + 945*a**4*b**3*d**2*e**5 -
2100*a**3*b**4*d**3*e**4 + 2205*a**2*b**5*d**4*e**3 - 1134*a*b**6*d**5*e**2 + 231*b**7*d**6*e))/(6*d**3*e**8
+ 18*d**2*e**9*x + 18*d*e**10*x**2 + 6*e**11*x**3)

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