Optimal. Leaf size=106 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B)}{6 (d+e x)^6 (b d-a e)^2}+\frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2} (B d-A e)}{7 (d+e x)^7 (b d-a e)^2} \]
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Rubi [A] time = 0.0596924, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {769, 646, 37} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B)}{6 (d+e x)^6 (b d-a e)^2}+\frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2} (B d-A e)}{7 (d+e x)^7 (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 769
Rule 646
Rule 37
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx &=\frac{(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (b d-a e)^2 (d+e x)^7}+\frac{(A b-a B) \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx}{b d-a e}\\ &=\frac{(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (b d-a e)^2 (d+e x)^7}+\frac{\left ((A b-a B) \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^7} \, dx}{b^4 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac{(A b-a B) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (b d-a e)^2 (d+e x)^6}+\frac{(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (b d-a e)^2 (d+e x)^7}\\ \end{align*}
Mathematica [B] time = 0.219265, size = 465, normalized size = 4.39 \[ -\frac{\sqrt{(a+b x)^2} \left (a^2 b^3 e^2 \left (3 A e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )+a^3 b^2 e^3 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )+a^4 b e^4 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+a^5 e^5 (6 A e+B (d+7 e x))+a b^4 e \left (2 A e \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+5 B \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )\right )+b^5 \left (A e \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )+6 B \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )\right )\right )}{42 e^7 (a+b x) (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 687, normalized size = 6.5 \begin{align*} -{\frac{42\,B{x}^{6}{b}^{5}{e}^{6}+21\,A{x}^{5}{b}^{5}{e}^{6}+105\,B{x}^{5}a{b}^{4}{e}^{6}+126\,B{x}^{5}{b}^{5}d{e}^{5}+70\,A{x}^{4}a{b}^{4}{e}^{6}+35\,A{x}^{4}{b}^{5}d{e}^{5}+140\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+175\,B{x}^{4}a{b}^{4}d{e}^{5}+210\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+105\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+70\,A{x}^{3}a{b}^{4}d{e}^{5}+35\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+105\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+140\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+175\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+210\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+84\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+63\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+42\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+21\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+42\,B{x}^{2}{a}^{4}b{e}^{6}+63\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+84\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+105\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+126\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+35\,Ax{a}^{4}b{e}^{6}+28\,Ax{a}^{3}{b}^{2}d{e}^{5}+21\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+14\,Axa{b}^{4}{d}^{3}{e}^{3}+7\,Ax{b}^{5}{d}^{4}{e}^{2}+7\,Bx{a}^{5}{e}^{6}+14\,Bx{a}^{4}bd{e}^{5}+21\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+28\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+35\,Bxa{b}^{4}{d}^{4}{e}^{2}+42\,Bx{b}^{5}{d}^{5}e+6\,A{a}^{5}{e}^{6}+5\,Ad{e}^{5}{a}^{4}b+4\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+3\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+2\,Aa{b}^{4}{d}^{4}{e}^{2}+A{b}^{5}{d}^{5}e+Bd{e}^{5}{a}^{5}+2\,B{a}^{4}b{d}^{2}{e}^{4}+3\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+4\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+5\,Ba{b}^{4}{d}^{5}e+6\,B{b}^{5}{d}^{6}}{42\, \left ( ex+d \right ) ^{7}{e}^{7} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58177, size = 1273, normalized size = 12.01 \begin{align*} -\frac{42 \, B b^{5} e^{6} x^{6} + 6 \, B b^{5} d^{6} + 6 \, A a^{5} e^{6} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 2 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 3 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 2 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} +{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 21 \,{\left (6 \, B b^{5} d e^{5} +{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 35 \,{\left (6 \, B b^{5} d^{2} e^{4} +{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 2 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 35 \,{\left (6 \, B b^{5} d^{3} e^{3} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 2 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 3 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 21 \,{\left (6 \, B b^{5} d^{4} e^{2} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 2 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 3 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 2 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 7 \,{\left (6 \, B b^{5} d^{5} e +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 2 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 3 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 2 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} +{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{42 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\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 = 1.18607, size = 1238, normalized size = 11.68 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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