Optimal. Leaf size=231 \[ f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 f g p \log \left (d+e x^2\right )}{2 e^2}-\frac{2 d^2 g^2 p x^3}{21 e^2}+\frac{2 d^3 g^2 p x}{7 e^3}-\frac{2 d^{7/2} g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}+\frac{2 \sqrt{d} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{d f g p x^2}{2 e}+\frac{2 d g^2 p x^5}{35 e}-2 f^2 p x-\frac{1}{4} f g p x^4-\frac{2}{49} g^2 p x^7 \]
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Rubi [A] time = 0.176745, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {2471, 2448, 321, 205, 2454, 2395, 43, 2455, 302} \[ f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{d^2 f g p \log \left (d+e x^2\right )}{2 e^2}-\frac{2 d^2 g^2 p x^3}{21 e^2}+\frac{2 d^3 g^2 p x}{7 e^3}-\frac{2 d^{7/2} g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}+\frac{2 \sqrt{d} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{d f g p x^2}{2 e}+\frac{2 d g^2 p x^5}{35 e}-2 f^2 p x-\frac{1}{4} f g p x^4-\frac{2}{49} g^2 p x^7 \]
Antiderivative was successfully verified.
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Rule 2471
Rule 2448
Rule 321
Rule 205
Rule 2454
Rule 2395
Rule 43
Rule 2455
Rule 302
Rubi steps
\begin{align*} \int \left (f+g x^3\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^2 \log \left (c \left (d+e x^2\right )^p\right )+2 f g x^3 \log \left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^2 \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+(2 f g) \int x^3 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+(f g) \operatorname{Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )-\left (2 e f^2 p\right ) \int \frac{x^2}{d+e x^2} \, dx-\frac{1}{7} \left (2 e g^2 p\right ) \int \frac{x^8}{d+e x^2} \, dx\\ &=-2 f^2 p x+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\left (2 d f^2 p\right ) \int \frac{1}{d+e x^2} \, dx-\frac{1}{2} (e f g p) \operatorname{Subst}\left (\int \frac{x^2}{d+e x} \, dx,x,x^2\right )-\frac{1}{7} \left (2 e g^2 p\right ) \int \left (-\frac{d^3}{e^4}+\frac{d^2 x^2}{e^3}-\frac{d x^4}{e^2}+\frac{x^6}{e}+\frac{d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=-2 f^2 p x+\frac{2 d^3 g^2 p x}{7 e^3}-\frac{2 d^2 g^2 p x^3}{21 e^2}+\frac{2 d g^2 p x^5}{35 e}-\frac{2}{49} g^2 p x^7+\frac{2 \sqrt{d} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{1}{2} (e f g p) \operatorname{Subst}\left (\int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx,x,x^2\right )-\frac{\left (2 d^4 g^2 p\right ) \int \frac{1}{d+e x^2} \, dx}{7 e^3}\\ &=-2 f^2 p x+\frac{2 d^3 g^2 p x}{7 e^3}+\frac{d f g p x^2}{2 e}-\frac{2 d^2 g^2 p x^3}{21 e^2}-\frac{1}{4} f g p x^4+\frac{2 d g^2 p x^5}{35 e}-\frac{2}{49} g^2 p x^7+\frac{2 \sqrt{d} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{2 d^{7/2} g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}-\frac{d^2 f g p \log \left (d+e x^2\right )}{2 e^2}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )\\ \end{align*}
Mathematica [A] time = 0.207592, size = 178, normalized size = 0.77 \[ \frac{1}{14} x \left (14 f^2+7 f g x^3+2 g^2 x^6\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac{p x \left (-280 d^2 e g^2 x^2+840 d^3 g^2+42 d e^2 g x \left (35 f+4 g x^3\right )-15 e^3 \left (392 f^2+49 f g x^3+8 g^2 x^6\right )\right )}{2940 e^3}-\frac{2 \sqrt{d} p \left (d^3 g^2-7 e^3 f^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}-\frac{d^2 f g p \log \left (d+e x^2\right )}{2 e^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.723, size = 869, normalized size = 3.8 \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 [A] time = 2.02251, size = 1027, normalized size = 4.45 \begin{align*} \left [-\frac{120 \, e^{3} g^{2} p x^{7} - 168 \, d e^{2} g^{2} p x^{5} + 735 \, e^{3} f g p x^{4} + 280 \, d^{2} e g^{2} p x^{3} - 1470 \, d e^{2} f g p x^{2} + 420 \,{\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p \sqrt{-\frac{d}{e}} \log \left (\frac{e x^{2} - 2 \, e x \sqrt{-\frac{d}{e}} - d}{e x^{2} + d}\right ) + 840 \,{\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p x - 210 \,{\left (2 \, e^{3} g^{2} p x^{7} + 7 \, e^{3} f g p x^{4} + 14 \, e^{3} f^{2} p x - 7 \, d^{2} e f g p\right )} \log \left (e x^{2} + d\right ) - 210 \,{\left (2 \, e^{3} g^{2} x^{7} + 7 \, e^{3} f g x^{4} + 14 \, e^{3} f^{2} x\right )} \log \left (c\right )}{2940 \, e^{3}}, -\frac{120 \, e^{3} g^{2} p x^{7} - 168 \, d e^{2} g^{2} p x^{5} + 735 \, e^{3} f g p x^{4} + 280 \, d^{2} e g^{2} p x^{3} - 1470 \, d e^{2} f g p x^{2} - 840 \,{\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p \sqrt{\frac{d}{e}} \arctan \left (\frac{e x \sqrt{\frac{d}{e}}}{d}\right ) + 840 \,{\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p x - 210 \,{\left (2 \, e^{3} g^{2} p x^{7} + 7 \, e^{3} f g p x^{4} + 14 \, e^{3} f^{2} p x - 7 \, d^{2} e f g p\right )} \log \left (e x^{2} + d\right ) - 210 \,{\left (2 \, e^{3} g^{2} x^{7} + 7 \, e^{3} f g x^{4} + 14 \, e^{3} f^{2} x\right )} \log \left (c\right )}{2940 \, e^{3}}\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 [A] time = 1.22128, size = 304, normalized size = 1.32 \begin{align*} -\frac{1}{2} \, d^{2} f g p e^{\left (-2\right )} \log \left (x^{2} e + d\right ) - \frac{2 \,{\left (d^{4} g^{2} p - 7 \, d f^{2} p e^{3}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{7}{2}\right )}}{7 \, \sqrt{d}} + \frac{1}{2940} \,{\left (420 \, g^{2} p x^{7} e^{3} \log \left (x^{2} e + d\right ) - 120 \, g^{2} p x^{7} e^{3} + 420 \, g^{2} x^{7} e^{3} \log \left (c\right ) + 168 \, d g^{2} p x^{5} e^{2} - 280 \, d^{2} g^{2} p x^{3} e + 1470 \, f g p x^{4} e^{3} \log \left (x^{2} e + d\right ) - 735 \, f g p x^{4} e^{3} + 1470 \, f g x^{4} e^{3} \log \left (c\right ) + 840 \, d^{3} g^{2} p x + 1470 \, d f g p x^{2} e^{2} + 2940 \, f^{2} p x e^{3} \log \left (x^{2} e + d\right ) - 5880 \, f^{2} p x e^{3} + 2940 \, f^{2} x e^{3} \log \left (c\right )\right )} e^{\left (-3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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