Optimal. Leaf size=304 \[ -\frac{(d+e x)^2 \left (2 a e \left (2 a A e^2+5 a B d e+A c d^2\right )-x \left (A c d \left (5 a e^2+3 c d^2\right )+5 a B e \left (c d^2-a e^2\right )\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{e^2 x \left (A c d \left (7 a e^2+3 c d^2\right )+5 a B e \left (c d^2-3 a e^2\right )\right )}{8 a^2 c^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+5 a B e \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^{5/2} c^{7/2}}+\frac{e^4 \log \left (a+c x^2\right ) (A e+5 B d)}{2 c^3}-\frac{(d+e x)^4 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.421686, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {819, 774, 635, 205, 260} \[ -\frac{(d+e x)^2 \left (2 a e \left (2 a A e^2+5 a B d e+A c d^2\right )-x \left (A c d \left (5 a e^2+3 c d^2\right )+5 a B e \left (c d^2-a e^2\right )\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{e^2 x \left (A c d \left (7 a e^2+3 c d^2\right )+5 a B e \left (c d^2-3 a e^2\right )\right )}{8 a^2 c^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+5 a B e \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^{5/2} c^{7/2}}+\frac{e^4 \log \left (a+c x^2\right ) (A e+5 B d)}{2 c^3}-\frac{(d+e x)^4 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 819
Rule 774
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^5}{\left (a+c x^2\right )^3} \, dx &=-\frac{(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}+\frac{\int \frac{(d+e x)^3 \left (3 A c d^2+a e (5 B d+4 A e)-e (A c d-5 a B e) x\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac{(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac{(d+e x)^2 \left (2 a e \left (A c d^2+5 a B d e+2 a A e^2\right )-\left (5 a B e \left (c d^2-a e^2\right )+A c d \left (3 c d^2+5 a e^2\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\int \frac{(d+e x) \left (5 a B d e \left (c d^2+5 a e^2\right )+A \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )-e \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right ) x\right )}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac{e^2 \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right ) x}{8 a^2 c^3}-\frac{(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac{(d+e x)^2 \left (2 a e \left (A c d^2+5 a B d e+2 a A e^2\right )-\left (5 a B e \left (c d^2-a e^2\right )+A c d \left (3 c d^2+5 a e^2\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\int \frac{a e^2 \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right )+c d \left (5 a B d e \left (c d^2+5 a e^2\right )+A \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )\right )+c \left (-d e \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right )+e \left (5 a B d e \left (c d^2+5 a e^2\right )+A \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )\right )\right ) x}{a+c x^2} \, dx}{8 a^2 c^3}\\ &=-\frac{e^2 \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right ) x}{8 a^2 c^3}-\frac{(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac{(d+e x)^2 \left (2 a e \left (A c d^2+5 a B d e+2 a A e^2\right )-\left (5 a B e \left (c d^2-a e^2\right )+A c d \left (3 c d^2+5 a e^2\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\left (e^4 (5 B d+A e)\right ) \int \frac{x}{a+c x^2} \, dx}{c^2}+\frac{\left (5 a B e \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+A c d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right )\right ) \int \frac{1}{a+c x^2} \, dx}{8 a^2 c^3}\\ &=-\frac{e^2 \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right ) x}{8 a^2 c^3}-\frac{(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac{(d+e x)^2 \left (2 a e \left (A c d^2+5 a B d e+2 a A e^2\right )-\left (5 a B e \left (c d^2-a e^2\right )+A c d \left (3 c d^2+5 a e^2\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\left (5 a B e \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+A c d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{7/2}}+\frac{e^4 (5 B d+A e) \log \left (a+c x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.288852, size = 341, normalized size = 1.12 \[ \frac{\frac{2 \sqrt{c} \left (5 a^2 c d e^2 (A e (2 d+e x)+2 B d (d+e x))-a^3 e^4 (A e+5 B d+B e x)-a c^2 d^3 (5 A e (d+2 e x)+B d (d+5 e x))+A c^3 d^5 x\right )}{a \left (a+c x^2\right )^2}+\frac{\sqrt{c} \left (-5 a^2 c d e^2 (A e (8 d+5 e x)+2 B d (4 d+5 e x))+a^3 e^4 (8 A e+40 B d+9 B e x)+5 a c^2 d^3 e x (2 A e+B d)+3 A c^3 d^5 x\right )}{a^2 \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+5 a B e \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )\right )}{a^{5/2}}+4 \sqrt{c} e^4 \log \left (a+c x^2\right ) (A e+5 B d)+8 B \sqrt{c} e^5 x}{8 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 678, normalized size = 2.2 \begin{align*} \text{result too large to display} \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 = 2.33834, size = 2911, normalized size = 9.58 \begin{align*} \text{result too large to display} \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 [A] time = 1.12251, size = 544, normalized size = 1.79 \begin{align*} \frac{B x e^{5}}{c^{3}} + \frac{{\left (5 \, B d e^{4} + A e^{5}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{{\left (3 \, A c^{3} d^{5} + 5 \, B a c^{2} d^{4} e + 10 \, A a c^{2} d^{3} e^{2} + 30 \, B a^{2} c d^{2} e^{3} + 15 \, A a^{2} c d e^{4} - 15 \, B a^{3} e^{5}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c^{3}} - \frac{2 \, B a^{2} c^{2} d^{5} + 10 \, A a^{2} c^{2} d^{4} e + 20 \, B a^{3} c d^{3} e^{2} + 20 \, A a^{3} c d^{2} e^{3} - 30 \, B a^{4} d e^{4} - 6 \, A a^{4} e^{5} -{\left (3 \, A c^{4} d^{5} + 5 \, B a c^{3} d^{4} e + 10 \, A a c^{3} d^{3} e^{2} - 50 \, B a^{2} c^{2} d^{2} e^{3} - 25 \, A a^{2} c^{2} d e^{4} + 9 \, B a^{3} c e^{5}\right )} x^{3} + 8 \,{\left (5 \, B a^{2} c^{2} d^{3} e^{2} + 5 \, A a^{2} c^{2} d^{2} e^{3} - 5 \, B a^{3} c d e^{4} - A a^{3} c e^{5}\right )} x^{2} -{\left (5 \, A a c^{3} d^{5} - 5 \, B a^{2} c^{2} d^{4} e - 10 \, A a^{2} c^{2} d^{3} e^{2} - 30 \, B a^{3} c d^{2} e^{3} - 15 \, A a^{3} c d e^{4} + 7 \, B a^{4} e^{5}\right )} x}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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