Optimal. Leaf size=209 \[ -\frac{(d+e x) \left (a e (3 a B e+5 a C d+3 A c d)-x \left (3 A c^2 d^2-a \left (4 a C e^2-c d (3 B e+C d)\right )\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (c d^2 (3 a B e+a C d+3 A c d)+3 a e^2 (a B e+3 a C d+A c d)\right )}{8 a^{5/2} c^{5/2}}-\frac{(d+e x)^3 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2}+\frac{C e^3 \log \left (a+c x^2\right )}{2 c^3} \]
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Rubi [A] time = 0.304423, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1645, 819, 635, 205, 260} \[ -\frac{(d+e x) \left (a e (3 a B e+5 a C d+3 A c d)-x \left (3 A c^2 d^2-a \left (4 a C e^2-c d (3 B e+C d)\right )\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (c d^2 (3 a B e+a C d+3 A c d)+3 a e^2 (a B e+3 a C d+A c d)\right )}{8 a^{5/2} c^{5/2}}-\frac{(d+e x)^3 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2}+\frac{C e^3 \log \left (a+c x^2\right )}{2 c^3} \]
Antiderivative was successfully verified.
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Rule 1645
Rule 819
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx &=-\frac{(a B-(A c-a C) x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac{\int \frac{(d+e x)^2 (-3 A c d-a C d-3 a B e-4 a C e x)}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac{(d+e x) \left (a e (3 A c d+5 a C d+3 a B e)-\left (3 A c^2 d^2-a \left (4 a C e^2-c d (C d+3 B e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{\int \frac{-3 a e^2 (A c d+3 a C d+a B e)-c d^2 (3 A c d+a C d+3 a B e)-8 a^2 C e^3 x}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac{(d+e x) \left (a e (3 A c d+5 a C d+3 a B e)-\left (3 A c^2 d^2-a \left (4 a C e^2-c d (C d+3 B e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\left (C e^3\right ) \int \frac{x}{a+c x^2} \, dx}{c^2}+\frac{\left (3 a e^2 (A c d+3 a C d+a B e)+c d^2 (3 A c d+a C d+3 a B e)\right ) \int \frac{1}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac{(d+e x) \left (a e (3 A c d+5 a C d+3 a B e)-\left (3 A c^2 d^2-a \left (4 a C e^2-c d (C d+3 B e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\left (3 a e^2 (A c d+3 a C d+a B e)+c d^2 (3 A c d+a C d+3 a B e)\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}+\frac{C e^3 \log \left (a+c x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.324269, size = 281, normalized size = 1.34 \[ \frac{\frac{2 a^2 c e (e (A e+3 B d+B e x)+3 C d (d+e x))-2 a^3 C e^3-2 a c^2 d \left (3 A e (d+e x)+B d (d+3 e x)+C d^2 x\right )+2 A c^3 d^3 x}{a \left (a+c x^2\right )^2}+\frac{-a^2 c e (e (4 A e+12 B d+5 B e x)+3 C d (4 d+5 e x))+8 a^3 C e^3+a c^2 d x \left (3 e (A e+B d)+C d^2\right )+3 A c^3 d^3 x}{a^2 \left (a+c x^2\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (3 A c d \left (a e^2+c d^2\right )+a \left (3 a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right )\right )}{a^{5/2}}+4 C e^3 \log \left (a+c x^2\right )}{8 c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 402, normalized size = 1.9 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( 3\,Acd{e}^{2}a+3\,A{d}^{3}{c}^{2}-5\,{a}^{2}B{e}^{3}+3\,Bc{d}^{2}ae-15\,C{a}^{2}d{e}^{2}+Cac{d}^{3} \right ){x}^{3}}{8\,{a}^{2}c}}-{\frac{e \left ( Ac{e}^{2}+3\,Bcde-2\,aC{e}^{2}+3\,Cc{d}^{2} \right ){x}^{2}}{2\,{c}^{2}}}-{\frac{ \left ( 3\,Acd{e}^{2}a-5\,A{d}^{3}{c}^{2}+3\,{a}^{2}B{e}^{3}+3\,Bc{d}^{2}ae+9\,C{a}^{2}d{e}^{2}+Cac{d}^{3} \right ) x}{8\,a{c}^{2}}}-{\frac{aA{e}^{3}c+3\,A{c}^{2}{d}^{2}e+3\,aBd{e}^{2}c+B{c}^{2}{d}^{3}-3\,{a}^{2}C{e}^{3}+3\,Cac{d}^{2}e}{4\,{c}^{3}}} \right ) }+{\frac{C{e}^{3}\ln \left ( c{x}^{2}+a \right ) }{2\,{c}^{3}}}+{\frac{3\,Ad{e}^{2}}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,A{d}^{3}}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,B{e}^{3}}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,B{d}^{2}e}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{9\,Cd{e}^{2}}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{C{d}^{3}}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 2.66185, size = 2325, normalized size = 11.12 \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 = 1.16451, size = 470, normalized size = 2.25 \begin{align*} \frac{C e^{3} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{{\left (C a c d^{3} + 3 \, A c^{2} d^{3} + 3 \, B a c d^{2} e + 9 \, C a^{2} d e^{2} + 3 \, A a c d e^{2} + 3 \, B a^{2} e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c^{2}} + \frac{{\left (C a c^{2} d^{3} + 3 \, A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e - 15 \, C a^{2} c d e^{2} + 3 \, A a c^{2} d e^{2} - 5 \, B a^{2} c e^{3}\right )} x^{3} - 4 \,{\left (3 \, C a^{2} c d^{2} e + 3 \, B a^{2} c d e^{2} - 2 \, C a^{3} e^{3} + A a^{2} c e^{3}\right )} x^{2} -{\left (C a^{2} c d^{3} - 5 \, A a c^{2} d^{3} + 3 \, B a^{2} c d^{2} e + 9 \, C a^{3} d e^{2} + 3 \, A a^{2} c d e^{2} + 3 \, B a^{3} e^{3}\right )} x - \frac{2 \,{\left (B a^{2} c^{2} d^{3} + 3 \, C a^{3} c d^{2} e + 3 \, A a^{2} c^{2} d^{2} e + 3 \, B a^{3} c d e^{2} - 3 \, C a^{4} e^{3} + A a^{3} c e^{3}\right )}}{c}}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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