Optimal. Leaf size=216 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )-a \left (3 a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right )\right )}{2 a^{3/2} c^{5/2}}-\frac{e \log \left (a+c x^2\right ) \left (2 a C e^2-c \left (e (A e+3 B d)+3 C d^2\right )\right )}{2 c^3}-\frac{3 e^2 x (A c d-a (B e+3 C d))}{2 a c^2}-\frac{(d+e x)^3 (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}-\frac{e^3 x^2 (A c-2 a C)}{2 a c^2} \]
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Rubi [A] time = 0.504723, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1645, 801, 635, 205, 260} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )-a \left (3 a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right )\right )}{2 a^{3/2} c^{5/2}}-\frac{e \log \left (a+c x^2\right ) \left (2 a C e^2-c \left (e (A e+3 B d)+3 C d^2\right )\right )}{2 c^3}-\frac{3 e^2 x (A c d-a (B e+3 C d))}{2 a c^2}-\frac{(d+e x)^3 (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}-\frac{e^3 x^2 (A c-2 a C)}{2 a c^2} \]
Antiderivative was successfully verified.
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Rule 1645
Rule 801
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 )^2} \, dx &=-\frac{(a B-(A c-a C) x) (d+e x)^3}{2 a c \left (a+c x^2\right )}-\frac{\int \frac{(d+e x)^2 (-A c d-a C d-3 a B e+2 (A c-2 a C) e x)}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)^3}{2 a c \left (a+c x^2\right )}-\frac{\int \left (\frac{3 e^2 (A c d-3 a C d-a B e)}{c}+\frac{2 (A c-2 a C) e^3 x}{c}-\frac{A c d \left (c d^2+3 a e^2\right )-a \left (3 a e^2 (3 C d+B e)-c d^2 (C d+3 B e)\right )-2 a e \left (2 a C e^2-c \left (3 C d^2+e (3 B d+A e)\right )\right ) x}{c \left (a+c x^2\right )}\right ) \, dx}{2 a c}\\ &=-\frac{3 e^2 (A c d-a (3 C d+B e)) x}{2 a c^2}-\frac{(A c-2 a C) e^3 x^2}{2 a c^2}-\frac{(a B-(A c-a C) x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac{\int \frac{A c d \left (c d^2+3 a e^2\right )-a \left (3 a e^2 (3 C d+B e)-c d^2 (C d+3 B e)\right )-2 a e \left (2 a C e^2-c \left (3 C d^2+e (3 B d+A e)\right )\right ) x}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac{3 e^2 (A c d-a (3 C d+B e)) x}{2 a c^2}-\frac{(A c-2 a C) e^3 x^2}{2 a c^2}-\frac{(a B-(A c-a C) x) (d+e x)^3}{2 a c \left (a+c x^2\right )}-\frac{\left (e \left (2 a C e^2-c \left (3 C d^2+e (3 B d+A e)\right )\right )\right ) \int \frac{x}{a+c x^2} \, dx}{c^2}+\frac{\left (A c d \left (c d^2+3 a e^2\right )-a \left (3 a e^2 (3 C d+B e)-c d^2 (C d+3 B e)\right )\right ) \int \frac{1}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac{3 e^2 (A c d-a (3 C d+B e)) x}{2 a c^2}-\frac{(A c-2 a C) e^3 x^2}{2 a c^2}-\frac{(a B-(A c-a C) x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac{\left (A c d \left (c d^2+3 a e^2\right )-a \left (3 a e^2 (3 C d+B e)-c d^2 (C d+3 B e)\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{5/2}}-\frac{e \left (2 a C e^2-c \left (3 C d^2+e (3 B d+A e)\right )\right ) \log \left (a+c x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.21356, size = 233, normalized size = 1.08 \[ \frac{\frac{a^2 c e (e (A e+3 B d+B e x)+3 C d (d+e x))-a^3 C e^3-a c^2 d \left (3 A e (d+e x)+B d (d+3 e x)+C d^2 x\right )+A c^3 d^3 x}{a \left (a+c x^2\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a \left (c d^2 (3 B e+C d)-3 a e^2 (B e+3 C d)\right )\right )}{a^{3/2}}+e \log \left (a+c x^2\right ) \left (-2 a C e^2+c e (A e+3 B d)+3 c C d^2\right )+2 c e^2 x (B e+3 C d)+c C e^3 x^2}{2 c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 484, normalized size = 2.2 \begin{align*}{\frac{aA{e}^{3}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{3\,A{d}^{2}e}{2\,c \left ( c{x}^{2}+a \right ) }}-{\frac{C{a}^{2}{e}^{3}}{2\,{c}^{3} \left ( c{x}^{2}+a \right ) }}+{\frac{A{d}^{3}}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{C{d}^{3}}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,\ln \left ( c{x}^{2}+a \right ) Bd{e}^{2}}{2\,{c}^{2}}}-{\frac{a\ln \left ( c{x}^{2}+a \right ) C{e}^{3}}{{c}^{3}}}+{\frac{xA{d}^{3}}{ \left ( 2\,c{x}^{2}+2\,a \right ) a}}+3\,{\frac{{e}^{2}Cdx}{{c}^{2}}}-{\frac{C{d}^{3}x}{2\,c \left ( c{x}^{2}+a \right ) }}+{\frac{3\,Cad{e}^{2}x}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{9\,Cad{e}^{2}}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{2}eaC}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{3\,Ba{e}^{3}}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,B{d}^{2}e}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,Ad{e}^{2}}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,\ln \left ( c{x}^{2}+a \right ) C{d}^{2}e}{2\,{c}^{2}}}-{\frac{3\,Ad{e}^{2}x}{2\,c \left ( c{x}^{2}+a \right ) }}+{\frac{Ba{e}^{3}x}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{3\,B{d}^{2}ex}{2\,c \left ( c{x}^{2}+a \right ) }}+{\frac{3\,aBd{e}^{2}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{C{e}^{3}{x}^{2}}{2\,{c}^{2}}}+{\frac{{e}^{3}Bx}{{c}^{2}}}-{\frac{B{d}^{3}}{2\,c \left ( c{x}^{2}+a \right ) }}+{\frac{\ln \left ( c{x}^{2}+a \right ) A{e}^{3}}{2\,{c}^{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 = 2.08102, size = 1881, normalized size = 8.71 \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 [B] time = 39.3074, size = 949, normalized size = 4.39 \begin{align*} \frac{C e^{3} x^{2}}{2 c^{2}} + \left (- \frac{e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} - \frac{\sqrt{- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right ) \log{\left (x + \frac{2 A a^{2} c e^{3} + 6 B a^{2} c d e^{2} - 4 C a^{3} e^{3} + 6 C a^{2} c d^{2} e - 4 a^{2} c^{3} \left (- \frac{e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} - \frac{\sqrt{- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right )}{- 3 A a c^{2} d e^{2} - A c^{3} d^{3} + 3 B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 9 C a^{2} c d e^{2} - C a c^{2} d^{3}} \right )} + \left (- \frac{e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} + \frac{\sqrt{- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right ) \log{\left (x + \frac{2 A a^{2} c e^{3} + 6 B a^{2} c d e^{2} - 4 C a^{3} e^{3} + 6 C a^{2} c d^{2} e - 4 a^{2} c^{3} \left (- \frac{e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} + \frac{\sqrt{- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right )}{- 3 A a c^{2} d e^{2} - A c^{3} d^{3} + 3 B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 9 C a^{2} c d e^{2} - C a c^{2} d^{3}} \right )} + \frac{A a^{2} c e^{3} - 3 A a c^{2} d^{2} e + 3 B a^{2} c d e^{2} - B a c^{2} d^{3} - C a^{3} e^{3} + 3 C a^{2} c d^{2} e + x \left (- 3 A a c^{2} d e^{2} + A c^{3} d^{3} + B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 3 C a^{2} c d e^{2} - C a c^{2} d^{3}\right )}{2 a^{2} c^{3} + 2 a c^{4} x^{2}} + \frac{x \left (B e^{3} + 3 C d e^{2}\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17587, size = 390, normalized size = 1.81 \begin{align*} \frac{{\left (3 \, C c d^{2} e + 3 \, B c d e^{2} - 2 \, C a e^{3} + A c e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{{\left (C a c d^{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 )}{2 \, \sqrt{a c} a c^{2}} + \frac{C c^{2} x^{2} e^{3} + 6 \, C c^{2} d x e^{2} + 2 \, B c^{2} x e^{3}}{2 \, c^{4}} - \frac{B a c^{2} d^{3} - 3 \, C a^{2} c d^{2} e + 3 \, A a c^{2} d^{2} e - 3 \, B a^{2} c d e^{2} + C a^{3} e^{3} - A a^{2} c e^{3} +{\left (C a c^{2} d^{3} - A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e - 3 \, C a^{2} c d e^{2} + 3 \, A a c^{2} d e^{2} - B a^{2} c e^{3}\right )} x}{2 \,{\left (c x^{2} + a\right )} a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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