Optimal. Leaf size=549 \[ -\frac{e}{\sqrt{-a} \sqrt{d+e x} \sqrt{f+g x} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g)}+\frac{e}{\sqrt{-a} \sqrt{d+e x} \sqrt{f+g x} \left (\sqrt{-a} e+\sqrt{c} d\right ) (e f-d g)}+\frac{g \sqrt{d+e x} \left (2 \sqrt{-a} e g-\sqrt{c} (d g+e f)\right )}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)^2}+\frac{g \sqrt{d+e x} \left (2 \sqrt{-a} e g+\sqrt{c} (d g+e f)\right )}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (\sqrt{-a} g+\sqrt{c} f\right ) (e f-d g)^2}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right )^{3/2} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \left (\sqrt{-a} e+\sqrt{c} d\right )^{3/2} \left (\sqrt{-a} g+\sqrt{c} f\right )^{3/2}} \]
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Rubi [A] time = 1.32362, antiderivative size = 543, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {912, 104, 152, 12, 93, 208} \[ -\frac{e}{\sqrt{-a} \sqrt{d+e x} \sqrt{f+g x} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g)}+\frac{e}{\sqrt{-a} \sqrt{d+e x} \sqrt{f+g x} \left (\sqrt{-a} e+\sqrt{c} d\right ) (e f-d g)}+\frac{g \sqrt{d+e x} \left (2 a e g-\sqrt{-a} \sqrt{c} (d g+e f)\right )}{a \sqrt{f+g x} \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (\sqrt{-a} g+\sqrt{c} f\right ) (e f-d g)^2}+\frac{g \sqrt{d+e x} \left (\sqrt{-a} \sqrt{c} (d g+e f)+2 a e g\right )}{a \sqrt{f+g x} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)^2}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right )^{3/2} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \left (\sqrt{-a} e+\sqrt{c} d\right )^{3/2} \left (\sqrt{-a} g+\sqrt{c} f\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 912
Rule 104
Rule 152
Rule 12
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{3/2} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx &=\int \left (\frac{\sqrt{-a}}{2 a \left (\sqrt{-a}-\sqrt{c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}}+\frac{\sqrt{-a}}{2 a \left (\sqrt{-a}+\sqrt{c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}}\right ) \, dx\\ &=-\frac{\int \frac{1}{\left (\sqrt{-a}-\sqrt{c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}} \, dx}{2 \sqrt{-a}}-\frac{\int \frac{1}{\left (\sqrt{-a}+\sqrt{c} x\right ) (d+e x)^{3/2} (f+g x)^{3/2}} \, dx}{2 \sqrt{-a}}\\ &=-\frac{e}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x} \sqrt{f+g x}}+\frac{e}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x} \sqrt{f+g x}}-\frac{\int \frac{\frac{1}{2} \left (2 \sqrt{-a} e g+\sqrt{c} (e f-d g)\right )+\sqrt{c} e g x}{\left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} (f+g x)^{3/2}} \, dx}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g)}+\frac{\int \frac{\frac{1}{2} \left (2 \sqrt{-a} e g-\sqrt{c} (e f-d g)\right )-\sqrt{c} e g x}{\left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} (f+g x)^{3/2}} \, dx}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) (e f-d g)}\\ &=-\frac{e}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x} \sqrt{f+g x}}+\frac{e}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x} \sqrt{f+g x}}+\frac{g \left (2 \sqrt{-a} e g-\sqrt{c} (e f+d g)\right ) \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)^2 \sqrt{f+g x}}+\frac{g \left (2 \sqrt{-a} e g+\sqrt{c} (e f+d g)\right ) \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{c} f+\sqrt{-a} g\right ) (e f-d g)^2 \sqrt{f+g x}}+\frac{2 \int -\frac{c (e f-d g)^2}{4 \left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)^2}+\frac{2 \int -\frac{c (e f-d g)^2}{4 \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{c} f+\sqrt{-a} g\right ) (e f-d g)^2}\\ &=-\frac{e}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x} \sqrt{f+g x}}+\frac{e}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x} \sqrt{f+g x}}+\frac{g \left (2 \sqrt{-a} e g-\sqrt{c} (e f+d g)\right ) \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)^2 \sqrt{f+g x}}+\frac{g \left (2 \sqrt{-a} e g+\sqrt{c} (e f+d g)\right ) \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{c} f+\sqrt{-a} g\right ) (e f-d g)^2 \sqrt{f+g x}}-\frac{c \int \frac{1}{\left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{c} f+\sqrt{-a} g\right )}-\frac{c \int \frac{1}{\left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \left (\sqrt{-a} c d f+(-a)^{3/2} e g+a \sqrt{c} (e f+d g)\right )}\\ &=-\frac{e}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x} \sqrt{f+g x}}+\frac{e}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x} \sqrt{f+g x}}+\frac{g \left (2 \sqrt{-a} e g-\sqrt{c} (e f+d g)\right ) \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)^2 \sqrt{f+g x}}+\frac{g \left (2 \sqrt{-a} e g+\sqrt{c} (e f+d g)\right ) \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{c} f+\sqrt{-a} g\right ) (e f-d g)^2 \sqrt{f+g x}}-\frac{c \operatorname{Subst}\left (\int \frac{1}{\sqrt{c} d+\sqrt{-a} e-\left (\sqrt{c} f+\sqrt{-a} g\right ) x^2} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{c} f+\sqrt{-a} g\right )}-\frac{c \operatorname{Subst}\left (\int \frac{1}{-\sqrt{c} d+\sqrt{-a} e-\left (-\sqrt{c} f+\sqrt{-a} g\right ) x^2} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{-a} c d f+(-a)^{3/2} e g+a \sqrt{c} (e f+d g)}\\ &=-\frac{e}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x} \sqrt{f+g x}}+\frac{e}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) (e f-d g) \sqrt{d+e x} \sqrt{f+g x}}+\frac{g \left (2 \sqrt{-a} e g-\sqrt{c} (e f+d g)\right ) \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)^2 \sqrt{f+g x}}+\frac{g \left (2 \sqrt{-a} e g+\sqrt{c} (e f+d g)\right ) \sqrt{d+e x}}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{c} f+\sqrt{-a} g\right ) (e f-d g)^2 \sqrt{f+g x}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} f-\sqrt{-a} g} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{f+g x}}\right )}{\sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (\sqrt{-a} c d f+(-a)^{3/2} e g+a \sqrt{c} (e f+d g)\right )}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} f+\sqrt{-a} g} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{-a} e} \sqrt{f+g x}}\right )}{\sqrt{-a} \left (\sqrt{c} d+\sqrt{-a} e\right )^{3/2} \left (\sqrt{c} f+\sqrt{-a} g\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 3.24467, size = 525, normalized size = 0.96 \[ \frac{\frac{e}{\sqrt{d+e x} \sqrt{f+g x} \left (\sqrt{-a} e-\sqrt{c} d\right )}+\frac{e}{\sqrt{d+e x} \sqrt{f+g x} \left (\sqrt{-a} e+\sqrt{c} d\right )}+\frac{g \sqrt{d+e x} \left (2 \sqrt{-a} e g+\sqrt{c} (d g+e f)\right )}{\sqrt{f+g x} \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (\sqrt{-a} g+\sqrt{c} f\right ) (e f-d g)}+\frac{c (e f-d g) \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{-\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\left (\sqrt{-a} e+\sqrt{c} d\right )^{3/2} \left (-\sqrt{-a} g-\sqrt{c} f\right )^{3/2}}+\frac{\frac{g \sqrt{d+e x} \left (2 \sqrt{-a} e g-\sqrt{c} (d g+e f)\right )}{\sqrt{f+g x} \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)}+\frac{c (d g-e f) \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right )}{\sqrt{\sqrt{-a} e-\sqrt{c} d} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}}{\sqrt{c} d-\sqrt{-a} e}}{\sqrt{-a} (e f-d g)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.652, size = 30648, normalized size = 55.8 \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{1}{{\left (c x^{2} + a\right )}{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{3}{2}}}\,{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{1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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