Optimal. Leaf size=149 \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{5/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{5/4}}-\frac{2 e \sqrt{d+e x}}{c} \]
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Rubi [A] time = 0.281862, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {704, 827, 1166, 208} \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{5/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{5/4}}-\frac{2 e \sqrt{d+e x}}{c} \]
Antiderivative was successfully verified.
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Rule 704
Rule 827
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{a-c x^2} \, dx &=-\frac{2 e \sqrt{d+e x}}{c}-\frac{\int \frac{-c d^2-a e^2-2 c d e x}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx}{c}\\ &=-\frac{2 e \sqrt{d+e x}}{c}-\frac{2 \operatorname{Subst}\left (\int \frac{2 c d^2 e+e \left (-c d^2-a e^2\right )-2 c d e x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt{d+e x}\right )}{c}\\ &=-\frac{2 e \sqrt{d+e x}}{c}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2 \operatorname{Subst}\left (\int \frac{1}{c d-\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{\sqrt{a} \sqrt{c}}+\frac{\left (\sqrt{c} d+\sqrt{a} e\right )^2 \operatorname{Subst}\left (\int \frac{1}{c d+\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{\sqrt{a} \sqrt{c}}\\ &=-\frac{2 e \sqrt{d+e x}}{c}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{5/4}}+\frac{\left (\sqrt{c} d+\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{a} e}}\right )}{\sqrt{a} c^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.115088, size = 147, normalized size = 0.99 \[ \frac{-2 \sqrt{a} \sqrt [4]{c} e \sqrt{d+e x}+\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (-\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )\right )+\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{5/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.209, size = 335, normalized size = 2.3 \begin{align*} -2\,{\frac{e\sqrt{ex+d}}{c}}+{a{e}^{3}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{ce{d}^{2}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-2\,{\frac{de}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) }+{a{e}^{3}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{ce{d}^{2}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+2\,{\frac{de}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}{\it Artanh} \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) } \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 (e x + d\right )}^{\frac{3}{2}}}{c x^{2} - a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03652, size = 1970, normalized size = 13.22 \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 = 37.8756, size = 316, normalized size = 2.12 \begin{align*} - \frac{2 a e^{3} \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} c e^{6} - 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} - 1, \left ( t \mapsto t \log{\left (- 64 t^{3} a^{2} c d e^{4} + 64 t^{3} a c^{2} d^{3} e^{2} - 4 t a e^{2} - 4 t c d^{2} + \sqrt{d + e x} \right )} \right )\right )}}{c} + 2 d^{2} e \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} c e^{6} - 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} - 1, \left ( t \mapsto t \log{\left (- 64 t^{3} a^{2} c d e^{4} + 64 t^{3} a c^{2} d^{3} e^{2} - 4 t a e^{2} - 4 t c d^{2} + \sqrt{d + e x} \right )} \right )\right )} - 4 d e \operatorname{RootSum}{\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left ( t \mapsto t \log{\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt{d + e x} \right )} \right )\right )} - \frac{2 e \sqrt{d + e x}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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