Optimal. Leaf size=496 \[ -\frac{\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (e-\sqrt{e^2-4 d f}\right ) \left (c^2 d e^2-f (c d-a f)^2\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (\sqrt{e^2-4 d f}+e\right ) \left (c^2 d e^2-f (c d-a f)^2\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d}-\frac{c^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{f^2}+\frac{\sqrt{a+c x^2} (c d-a f)}{d f}+\frac{a \sqrt{a+c x^2}}{d} \]
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Rubi [A] time = 2.56885, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {6728, 266, 50, 63, 208, 1020, 1080, 217, 206, 1034, 725} \[ -\frac{\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (e-\sqrt{e^2-4 d f}\right ) \left (c^2 d e^2-f (c d-a f)^2\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (\sqrt{e^2-4 d f}+e\right ) \left (c^2 d e^2-f (c d-a f)^2\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d}-\frac{c^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{f^2}+\frac{\sqrt{a+c x^2} (c d-a f)}{d f}+\frac{a \sqrt{a+c x^2}}{d} \]
Antiderivative was successfully verified.
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Rule 6728
Rule 266
Rule 50
Rule 63
Rule 208
Rule 1020
Rule 1080
Rule 217
Rule 206
Rule 1034
Rule 725
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^{3/2}}{x \left (d+e x+f x^2\right )} \, dx &=\int \left (\frac{\left (a+c x^2\right )^{3/2}}{d x}+\frac{(-e-f x) \left (a+c x^2\right )^{3/2}}{d \left (d+e x+f x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\left (a+c x^2\right )^{3/2}}{x} \, dx}{d}+\frac{\int \frac{(-e-f x) \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx}{d}\\ &=-\frac{\left (a+c x^2\right )^{3/2}}{3 d}+\frac{\operatorname{Subst}\left (\int \frac{(a+c x)^{3/2}}{x} \, dx,x,x^2\right )}{2 d}+\frac{\int \frac{(-3 a e f+3 f (c d-a f) x) \sqrt{a+c x^2}}{d+e x+f x^2} \, dx}{3 d f}\\ &=\frac{(c d-a f) \sqrt{a+c x^2}}{d f}+\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x} \, dx,x,x^2\right )}{2 d}+\frac{\int \frac{-3 a^2 e f^2-3 f (c d-a f)^2 x-3 c^2 d e f x^2}{\sqrt{a+c x^2} \left (d+e x+f x^2\right )} \, dx}{3 d f^2}\\ &=\frac{a \sqrt{a+c x^2}}{d}+\frac{(c d-a f) \sqrt{a+c x^2}}{d f}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{2 d}+\frac{\int \frac{3 c^2 d^2 e f-3 a^2 e f^3+\left (3 c^2 d e^2 f-3 f^2 (c d-a f)^2\right ) x}{\sqrt{a+c x^2} \left (d+e x+f x^2\right )} \, dx}{3 d f^3}-\frac{\left (c^2 e\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{f^2}\\ &=\frac{a \sqrt{a+c x^2}}{d}+\frac{(c d-a f) \sqrt{a+c x^2}}{d f}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{c d}-\frac{\left (c^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{f^2}+\frac{\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e-\sqrt{e^2-4 d f}\right )\right ) \int \frac{1}{\left (e-\sqrt{e^2-4 d f}+2 f x\right ) \sqrt{a+c x^2}} \, dx}{d f^2 \sqrt{e^2-4 d f}}-\frac{\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e+\sqrt{e^2-4 d f}\right )\right ) \int \frac{1}{\left (e+\sqrt{e^2-4 d f}+2 f x\right ) \sqrt{a+c x^2}} \, dx}{d f^2 \sqrt{e^2-4 d f}}\\ &=\frac{a \sqrt{a+c x^2}}{d}+\frac{(c d-a f) \sqrt{a+c x^2}}{d f}-\frac{c^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{f^2}-\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d}-\frac{\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e-\sqrt{e^2-4 d f}\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 )}{d f^2 \sqrt{e^2-4 d f}}+\frac{\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e+\sqrt{e^2-4 d f}\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 )}{d f^2 \sqrt{e^2-4 d f}}\\ &=\frac{a \sqrt{a+c x^2}}{d}+\frac{(c d-a f) \sqrt{a+c x^2}}{d f}-\frac{c^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{f^2}-\frac{\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e-\sqrt{e^2-4 d f}\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} d f^2 \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 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e+\sqrt{e^2-4 d f}\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} d f^2 \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{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d}\\ \end{align*}
Mathematica [A] time = 1.62658, size = 746, normalized size = 1.5 \[ -\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d}-\frac{c^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{f^2}+\frac{a \sqrt{a f^2+\frac{1}{2} c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{a+c x^2} \sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{2 d f}-\frac{a e \sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{a+c x^2} \sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{4 d f \sqrt{e^2-4 d f}}-\frac{\left (c d \left (\sqrt{e^2-4 d f}-e\right )-a f \left (\sqrt{e^2-4 d f}+e\right )\right ) \sqrt{4 a f^2-2 c \left (e \sqrt{e^2-4 d f}+2 d f-e^2\right )} \tanh ^{-1}\left (\frac{2 a f+c x \left (\sqrt{e^2-4 d f}-e\right )}{\sqrt{a+c x^2} \sqrt{4 a f^2-2 c \left (e \sqrt{e^2-4 d f}+2 d f-e^2\right )}}\right )}{4 d f^2 \sqrt{e^2-4 d f}}-\frac{c \sqrt{a f^2+\frac{1}{2} c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{a+c x^2} \sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{2 f^2}-\frac{c e \sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{a+c x^2} \sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{4 f^2 \sqrt{e^2-4 d f}}+\frac{c \sqrt{a+c x^2}}{f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.262, size = 9728, normalized size = 19.6 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}{{\left (f x^{2} + e x + d\right )} x}\,{d x} \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{\left (a + c x^{2}\right )^{\frac{3}{2}}}{x \left (d + e x + f x^{2}\right )}\, 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}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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