Optimal. Leaf size=396 \[ -\frac{\sqrt{e} \sqrt{c+d x^2} \left (5 a f (d e-3 c f)-b \left (4 d e^2-6 c e f\right )\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{15 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \left (10 a d f (d e-2 c f)-b \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )\right )}{15 d f^2 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left (10 a d f (d e-2 c f)-b \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 d f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (-5 a d f-3 b c f+4 b d e)}{15 f^2}+\frac{b x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 f} \]
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Rubi [A] time = 0.447957, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {528, 531, 418, 492, 411} \[ -\frac{x \sqrt{c+d x^2} \left (10 a d f (d e-2 c f)-b \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )\right )}{15 d f^2 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left (10 a d f (d e-2 c f)-b \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 d f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (5 a f (d e-3 c f)-b \left (4 d e^2-6 c e f\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (-5 a d f-3 b c f+4 b d e)}{15 f^2}+\frac{b x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 f} \]
Antiderivative was successfully verified.
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Rule 528
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}{\sqrt{e+f x^2}} \, dx &=\frac{b x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 f}+\frac{\int \frac{\sqrt{c+d x^2} \left (-c (b e-5 a f)+(-4 b d e+3 b c f+5 a d f) x^2\right )}{\sqrt{e+f x^2}} \, dx}{5 f}\\ &=-\frac{(4 b d e-3 b c f-5 a d f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 f^2}+\frac{b x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 f}+\frac{\int \frac{-c (5 a f (d e-3 c f)-2 b e (2 d e-3 c f))+\left (-10 a d f (d e-2 c f)+b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 f^2}\\ &=-\frac{(4 b d e-3 b c f-5 a d f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 f^2}+\frac{b x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 f}-\frac{\left (c \left (5 a f (d e-3 c f)-b \left (4 d e^2-6 c e f\right )\right )\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 f^2}-\frac{\left (10 a d f (d e-2 c f)-b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 f^2}\\ &=-\frac{\left (10 a d f (d e-2 c f)-b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x \sqrt{c+d x^2}}{15 d f^2 \sqrt{e+f x^2}}-\frac{(4 b d e-3 b c f-5 a d f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 f^2}+\frac{b x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 f}-\frac{\sqrt{e} \left (5 a f (d e-3 c f)-b \left (4 d e^2-6 c e f\right )\right ) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 f^{5/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{\left (e \left (10 a d f (d e-2 c f)-b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right )\right ) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{15 d f^2}\\ &=-\frac{\left (10 a d f (d e-2 c f)-b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x \sqrt{c+d x^2}}{15 d f^2 \sqrt{e+f x^2}}-\frac{(4 b d e-3 b c f-5 a d f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 f^2}+\frac{b x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 f}+\frac{\sqrt{e} \left (10 a d f (d e-2 c f)-b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 d f^{5/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}-\frac{\sqrt{e} \left (5 a f (d e-3 c f)-b \left (4 d e^2-6 c e f\right )\right ) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 f^{5/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}\\ \end{align*}
Mathematica [C] time = 0.82325, size = 279, normalized size = 0.7 \[ \frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (5 a f (2 d e-3 c f)+b e (9 c f-8 d e)) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (b \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-10 a d f (d e-2 c f)\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (5 a d f+b \left (6 c f-4 d e+3 d f x^2\right )\right )}{15 f^3 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 924, normalized size = 2.3 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{\sqrt{f x^{2} + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b d x^{4} +{\left (b c + a d\right )} x^{2} + a c\right )} \sqrt{d x^{2} + c}}{\sqrt{f x^{2} + e}}, x\right ) \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 + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}}}{\sqrt{e + 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*} \int \frac{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{\sqrt{f x^{2} + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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