Optimal. Leaf size=185 \[ \frac{b^2 (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{840 e (d+e x)^5 (b d-a e)^4}+\frac{b (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac{(a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac{(a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]
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Rubi [A] time = 0.0895438, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {27, 78, 45, 37} \[ \frac{b^2 (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{840 e (d+e x)^5 (b d-a e)^4}+\frac{b (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac{(a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac{(a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{(d+e x)^9} \, dx\\ &=-\frac{(B d-A e) (a+b x)^5}{8 e (b d-a e) (d+e x)^8}+\frac{(5 b B d+3 A b e-8 a B e) \int \frac{(a+b x)^4}{(d+e x)^8} \, dx}{8 e (b d-a e)}\\ &=-\frac{(B d-A e) (a+b x)^5}{8 e (b d-a e) (d+e x)^8}+\frac{(5 b B d+3 A b e-8 a B e) (a+b x)^5}{56 e (b d-a e)^2 (d+e x)^7}+\frac{(b (5 b B d+3 A b e-8 a B e)) \int \frac{(a+b x)^4}{(d+e x)^7} \, dx}{28 e (b d-a e)^2}\\ &=-\frac{(B d-A e) (a+b x)^5}{8 e (b d-a e) (d+e x)^8}+\frac{(5 b B d+3 A b e-8 a B e) (a+b x)^5}{56 e (b d-a e)^2 (d+e x)^7}+\frac{b (5 b B d+3 A b e-8 a B e) (a+b x)^5}{168 e (b d-a e)^3 (d+e x)^6}+\frac{\left (b^2 (5 b B d+3 A b e-8 a B e)\right ) \int \frac{(a+b x)^4}{(d+e x)^6} \, dx}{168 e (b d-a e)^3}\\ &=-\frac{(B d-A e) (a+b x)^5}{8 e (b d-a e) (d+e x)^8}+\frac{(5 b B d+3 A b e-8 a B e) (a+b x)^5}{56 e (b d-a e)^2 (d+e x)^7}+\frac{b (5 b B d+3 A b e-8 a B e) (a+b x)^5}{168 e (b d-a e)^3 (d+e x)^6}+\frac{b^2 (5 b B d+3 A b e-8 a B e) (a+b x)^5}{840 e (b d-a e)^4 (d+e x)^5}\\ \end{align*}
Mathematica [A] time = 0.146534, size = 320, normalized size = 1.73 \[ -\frac{6 a^2 b^2 e^2 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )\right )+20 a^3 b e^3 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+15 a^4 e^4 (7 A e+B (d+8 e x))+12 a b^3 e \left (A e \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+B \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )\right )+b^4 \left (3 A e \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )\right )}{840 e^6 (d+e x)^8} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 430, normalized size = 2.3 \begin{align*} -{\frac{A{a}^{4}{e}^{5}-4\,Ad{a}^{3}b{e}^{4}+6\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-4\,A{d}^{3}a{b}^{3}{e}^{2}+A{d}^{4}{b}^{4}e-Bd{a}^{4}{e}^{4}+4\,B{d}^{2}{a}^{3}b{e}^{3}-6\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+4\,B{d}^{4}a{b}^{3}e-{b}^{4}B{d}^{5}}{8\,{e}^{6} \left ( ex+d \right ) ^{8}}}-{\frac{{b}^{3} \left ( Abe+4\,aBe-5\,Bbd \right ) }{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{b \left ( 3\,A{a}^{2}b{e}^{3}-6\,Aa{b}^{2}d{e}^{2}+3\,A{b}^{3}{d}^{2}e+2\,B{e}^{3}{a}^{3}-9\,B{a}^{2}bd{e}^{2}+12\,Ba{b}^{2}{d}^{2}e-5\,B{b}^{3}{d}^{3} \right ) }{3\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{4\,A{a}^{3}b{e}^{4}-12\,Ad{a}^{2}{b}^{2}{e}^{3}+12\,A{d}^{2}a{b}^{3}{e}^{2}-4\,A{d}^{3}{b}^{4}e+B{e}^{4}{a}^{4}-8\,Bd{a}^{3}b{e}^{3}+18\,B{d}^{2}{a}^{2}{b}^{2}{e}^{2}-16\,B{d}^{3}a{b}^{3}e+5\,{b}^{4}B{d}^{4}}{7\,{e}^{6} \left ( ex+d \right ) ^{7}}}-{\frac{2\,{b}^{2} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+3\,{a}^{2}B{e}^{2}-8\,Bdabe+5\,B{b}^{2}{d}^{2} \right ) }{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{4}B}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16689, size = 660, normalized size = 3.57 \begin{align*} -\frac{280 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 105 \, A a^{4} e^{5} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 70 \,{\left (5 \, B b^{4} d e^{4} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 56 \,{\left (5 \, B b^{4} d^{2} e^{3} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 28 \,{\left (5 \, B b^{4} d^{3} e^{2} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 8 \,{\left (5 \, B b^{4} d^{4} e + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{840 \,{\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49133, size = 1034, normalized size = 5.59 \begin{align*} -\frac{280 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 105 \, A a^{4} e^{5} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 70 \,{\left (5 \, B b^{4} d e^{4} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 56 \,{\left (5 \, B b^{4} d^{2} e^{3} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 28 \,{\left (5 \, B b^{4} d^{3} e^{2} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 8 \,{\left (5 \, B b^{4} d^{4} e + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{840 \,{\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15232, size = 594, normalized size = 3.21 \begin{align*} -\frac{{\left (280 \, B b^{4} x^{5} e^{5} + 350 \, B b^{4} d x^{4} e^{4} + 280 \, B b^{4} d^{2} x^{3} e^{3} + 140 \, B b^{4} d^{3} x^{2} e^{2} + 40 \, B b^{4} d^{4} x e + 5 \, B b^{4} d^{5} + 840 \, B a b^{3} x^{4} e^{5} + 210 \, A b^{4} x^{4} e^{5} + 672 \, B a b^{3} d x^{3} e^{4} + 168 \, A b^{4} d x^{3} e^{4} + 336 \, B a b^{3} d^{2} x^{2} e^{3} + 84 \, A b^{4} d^{2} x^{2} e^{3} + 96 \, B a b^{3} d^{3} x e^{2} + 24 \, A b^{4} d^{3} x e^{2} + 12 \, B a b^{3} d^{4} e + 3 \, A b^{4} d^{4} e + 1008 \, B a^{2} b^{2} x^{3} e^{5} + 672 \, A a b^{3} x^{3} e^{5} + 504 \, B a^{2} b^{2} d x^{2} e^{4} + 336 \, A a b^{3} d x^{2} e^{4} + 144 \, B a^{2} b^{2} d^{2} x e^{3} + 96 \, A a b^{3} d^{2} x e^{3} + 18 \, B a^{2} b^{2} d^{3} e^{2} + 12 \, A a b^{3} d^{3} e^{2} + 560 \, B a^{3} b x^{2} e^{5} + 840 \, A a^{2} b^{2} x^{2} e^{5} + 160 \, B a^{3} b d x e^{4} + 240 \, A a^{2} b^{2} d x e^{4} + 20 \, B a^{3} b d^{2} e^{3} + 30 \, A a^{2} b^{2} d^{2} e^{3} + 120 \, B a^{4} x e^{5} + 480 \, A a^{3} b x e^{5} + 15 \, B a^{4} d e^{4} + 60 \, A a^{3} b d e^{4} + 105 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{840 \,{\left (x e + d\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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