Optimal. Leaf size=543 \[ \frac{e^{3/2} \sqrt{c+d x^2} \left (7 a d f (9 d e-c f)-b \left (-4 c^2 f^2+9 c d e f+3 d^2 e^2\right )\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{105 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (14 a d f (3 d e-c f)+b \left (8 c^2 f^2-15 c d e f+3 d^2 e^2\right )\right )}{105 d^2 f}+\frac{x \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right )\right )}{105 d^3 f \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (7 a d f-4 b c f+3 b d e)}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d} \]
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Rubi [A] time = 0.64488, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (14 a d f (3 d e-c f)+b \left (8 c^2 f^2-15 c d e f+3 d^2 e^2\right )\right )}{105 d^2 f}+\frac{x \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right )\right )}{105 d^3 f \sqrt{e+f x^2}}+\frac{e^{3/2} \sqrt{c+d x^2} \left (7 a d f (9 d e-c f)-b \left (-4 c^2 f^2+9 c d e f+3 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^2 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )-b \left (19 c^2 d e f^2-8 c^3 f^3-9 c d^2 e^2 f+6 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d^3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (7 a d f-4 b c f+3 b d e)}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 528
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \left (a+b x^2\right ) \sqrt{c+d x^2} \left (e+f x^2\right )^{3/2} \, dx &=\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac{\int \sqrt{c+d x^2} \sqrt{e+f x^2} \left (-(b c-7 a d) e+(3 b d e-4 b c f+7 a d f) x^2\right ) \, dx}{7 d}\\ &=\frac{(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac{\int \frac{\sqrt{c+d x^2} \left (-e (4 b c (2 d e-c f)-7 a d (5 d e-c f))+\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x^2\right )}{\sqrt{e+f x^2}} \, dx}{35 d^2}\\ &=\frac{\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d^2 f}+\frac{(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac{\int \frac{c e \left (7 a d f (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 c^2 f^2\right )\right )+\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 d^2 f}\\ &=\frac{\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d^2 f}+\frac{(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac{\left (c e \left (7 a d f (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 c^2 f^2\right )\right )\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 d^2 f}+\frac{\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 d^2 f}\\ &=\frac{\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) x \sqrt{c+d x^2}}{105 d^3 f \sqrt{e+f x^2}}+\frac{\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d^2 f}+\frac{(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}+\frac{e^{3/2} \left (7 a d f (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 c^2 f^2\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 )}{105 d^2 f^{3/2} \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 (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right )\right ) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{105 d^3 f}\\ &=\frac{\left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\right )\right ) x \sqrt{c+d x^2}}{105 d^3 f \sqrt{e+f x^2}}+\frac{\left (14 a d f (3 d e-c f)+b \left (3 d^2 e^2-15 c d e f+8 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 d^2 f}+\frac{(3 b d e-4 b c f+7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 d^2}+\frac{b x \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{7 d}-\frac{\sqrt{e} \left (7 a d f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b \left (6 d^3 e^3-9 c d^2 e^2 f+19 c^2 d e f^2-8 c^3 f^3\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 )}{105 d^3 f^{3/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{e^{3/2} \left (7 a d f (9 d e-c f)-b \left (3 d^2 e^2+9 c d e f-4 c^2 f^2\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 )}{105 d^2 f^{3/2} \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 = 1.14135, size = 372, normalized size = 0.69 \[ \frac{i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left (b \left (4 c^2 f^2-6 c d e f+6 d^2 e^2\right )-7 a d f (c f+3 d e)\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )+f x \left (-\sqrt{\frac{d}{c}}\right ) \left (c+d x^2\right ) \left (e+f x^2\right ) \left (-7 a d f \left (c f+6 d e+3 d f x^2\right )+4 b c^2 f^2-3 b c d f \left (3 e+f x^2\right )-3 b d^2 \left (e^2+8 e f x^2+5 f^2 x^4\right )\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (7 a d f \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )+b \left (-19 c^2 d e f^2+8 c^3 f^3+9 c d^2 e^2 f-6 d^3 e^3\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{105 c^2 f^2 \left (\frac{d}{c}\right )^{5/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 1332, normalized size = 2.5 \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{\left (b x^{2} + a\right )} \sqrt{d x^{2} + c}{\left (f x^{2} + e\right )}^{\frac{3}{2}}\,{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 ({\left (b f x^{4} +{\left (b e + a f\right )} x^{2} + a e\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 \left (a + b x^{2}\right ) \sqrt{c + d x^{2}} \left (e + f x^{2}\right )^{\frac{3}{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{\left (b x^{2} + a\right )} \sqrt{d x^{2} + c}{\left (f x^{2} + e\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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