Optimal. Leaf size=553 \[ -\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt{e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (\sqrt{e^2-4 d f}+e\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\sqrt{a+c x^2} \left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right )}{2 f^3}-\frac{\sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )}{2 f^4}+\frac{\left (a+c x^2\right )^{3/2}}{3 f} \]
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Rubi [A] time = 2.43059, antiderivative size = 553, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {1020, 1068, 1080, 217, 206, 1034, 725} \[ -\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt{e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (\sqrt{e^2-4 d f}+e\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\sqrt{a+c x^2} \left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right )}{2 f^3}-\frac{\sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )}{2 f^4}+\frac{\left (a+c x^2\right )^{3/2}}{3 f} \]
Antiderivative was successfully verified.
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Rule 1020
Rule 1068
Rule 1080
Rule 217
Rule 206
Rule 1034
Rule 725
Rubi steps
\begin{align*} \int \frac{x \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx &=\frac{\left (a+c x^2\right )^{3/2}}{3 f}+\frac{\int \frac{\sqrt{a+c x^2} \left (-3 (c d-a f) x-3 c e x^2\right )}{d+e x+f x^2} \, dx}{3 f}\\ &=\frac{\left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right ) \sqrt{a+c x^2}}{2 f^3}+\frac{\left (a+c x^2\right )^{3/2}}{3 f}-\frac{\int \frac{-3 a c^2 d e f+3 c \left (a c e^2 f+2 (c d-a f) \left (c e^2-c d f+a f^2\right )\right ) x+3 c^2 e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right ) x^2}{\sqrt{a+c x^2} \left (d+e x+f x^2\right )} \, dx}{6 c f^3}\\ &=\frac{\left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right ) \sqrt{a+c x^2}}{2 f^3}+\frac{\left (a+c x^2\right )^{3/2}}{3 f}-\frac{\int \frac{-3 a c^2 d e f^2-3 c^2 d e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )+\left (-3 c^2 e^2 \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )+3 c f \left (a c e^2 f+2 (c d-a f) \left (c e^2-c d f+a f^2\right )\right )\right ) x}{\sqrt{a+c x^2} \left (d+e x+f x^2\right )} \, dx}{6 c f^4}-\frac{\left (c e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{2 f^4}\\ &=\frac{\left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right ) \sqrt{a+c x^2}}{2 f^3}+\frac{\left (a+c x^2\right )^{3/2}}{3 f}-\frac{\left (c e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{2 f^4}+\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt{e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) \int \frac{1}{\left (e-\sqrt{e^2-4 d f}+2 f x\right ) \sqrt{a+c x^2}} \, dx}{f^4 \sqrt{e^2-4 d f}}-\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e+\sqrt{e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) \int \frac{1}{\left (e+\sqrt{e^2-4 d f}+2 f x\right ) \sqrt{a+c x^2}} \, dx}{f^4 \sqrt{e^2-4 d f}}\\ &=\frac{\left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right ) \sqrt{a+c x^2}}{2 f^3}+\frac{\left (a+c x^2\right )^{3/2}}{3 f}-\frac{\sqrt{c} e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 f^4}-\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt{e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a f^2+c \left (e-\sqrt{e^2-4 d f}\right )^2-x^2} \, dx,x,\frac{2 a f-c \left (e-\sqrt{e^2-4 d f}\right ) x}{\sqrt{a+c x^2}}\right )}{f^4 \sqrt{e^2-4 d f}}+\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e+\sqrt{e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a f^2+c \left (e+\sqrt{e^2-4 d f}\right )^2-x^2} \, dx,x,\frac{2 a f-c \left (e+\sqrt{e^2-4 d f}\right ) x}{\sqrt{a+c x^2}}\right )}{f^4 \sqrt{e^2-4 d f}}\\ &=\frac{\left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right ) \sqrt{a+c x^2}}{2 f^3}+\frac{\left (a+c x^2\right )^{3/2}}{3 f}-\frac{\sqrt{c} e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 f^4}-\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt{e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e-\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2-2 d f-e \sqrt{e^2-4 d f}\right )} \sqrt{a+c x^2}}\right )}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-2 d f-e \sqrt{e^2-4 d f}\right )}}+\frac{\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e+\sqrt{e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e+\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2-2 d f+e \sqrt{e^2-4 d f}\right )} \sqrt{a+c x^2}}\right )}{\sqrt{2} f^4 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-2 d f+e \sqrt{e^2-4 d f}\right )}}\\ \end{align*}
Mathematica [A] time = 2.16838, size = 755, normalized size = 1.37 \[ \frac{8 f^3 \left (a+c x^2\right )^{5/2} \sqrt{\frac{c x^2}{a}+1} \left (\sqrt{e^2-4 d f}-e\right )+8 f^3 \left (a+c x^2\right )^{5/2} \sqrt{\frac{c x^2}{a}+1} \left (\sqrt{e^2-4 d f}+e\right )+3 \left (e-\sqrt{e^2-4 d f}\right ) \left (2 \sqrt{c} f^2 \sqrt{a+c x^2} \left (e-\sqrt{e^2-4 d f}\right ) \left (a \sqrt{c} x \left (\frac{c x^2}{a}+1\right )^{3/2}+\sqrt{a} \left (a+c x^2\right ) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )\right )-a \left (\frac{c x^2}{a}+1\right )^{3/2} \left (4 a f^2+c \left (e-\sqrt{e^2-4 d f}\right )^2\right ) \left (-\sqrt{4 a f^2-2 c e \sqrt{e^2-4 d f}-4 c d f+2 c e^2} \tanh ^{-1}\left (\frac{2 a f+c x \left (\sqrt{e^2-4 d f}-e\right )}{\sqrt{a+c x^2} \sqrt{4 a f^2-2 c \left (e \sqrt{e^2-4 d f}+2 d f-e^2\right )}}\right )+\sqrt{c} \left (\sqrt{e^2-4 d f}-e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+2 f \sqrt{a+c x^2}\right )\right )-3 \left (\sqrt{e^2-4 d f}+e\right ) \left (2 \sqrt{c} f^2 \sqrt{a+c x^2} \left (\sqrt{e^2-4 d f}+e\right ) \left (a \sqrt{c} x \left (\frac{c x^2}{a}+1\right )^{3/2}+\sqrt{a} \left (a+c x^2\right ) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )\right )-a \left (\frac{c x^2}{a}+1\right )^{3/2} \left (4 a f^2+c \left (\sqrt{e^2-4 d f}+e\right )^2\right ) \left (-\sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{a+c x^2} \sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )-\sqrt{c} \left (\sqrt{e^2-4 d f}+e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+2 f \sqrt{a+c x^2}\right )\right )}{48 a f^4 \left (\frac{c x^2}{a}+1\right )^{3/2} \sqrt{e^2-4 d f}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.26, size = 14709, normalized size = 26.6 \begin{align*} \text{output 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 [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (a + c x^{2}\right )^{\frac{3}{2}}}{d + e x + f x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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