Optimal. Leaf size=531 \[ -\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} \left (3 a d \left (c^2 f^2-11 c d e f+8 d^2 e^2\right )+2 b c \left (2 c^2 f^2-c d e f+2 d^2 e^2\right )\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{105 c^4 d^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{e+f x^2} \left (3 a d \left (-2 c^2 f^2-5 c d e f+8 d^2 e^2\right )+b c \left (-8 c^2 f^2+c d e f+4 d^2 e^2\right )\right )}{105 c^3 d^2 \left (c+d x^2\right )^{3/2} (d e-c f)}+\frac{\sqrt{e+f x^2} \left (6 a d \left (2 c^2 d e f^2+c^3 f^3-12 c d^2 e^2 f+8 d^3 e^3\right )+b c \left (-5 c^2 d e f^2+8 c^3 f^3-5 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{105 c^{7/2} d^{5/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{x \sqrt{e+f x^2} (d e (6 a d+b c)-c f (3 a d+4 b c))}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}-\frac{x \left (e+f x^2\right )^{3/2} (b c-a d)}{7 c d \left (c+d x^2\right )^{7/2}} \]
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Rubi [A] time = 0.614756, antiderivative size = 531, 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 = {526, 527, 525, 418, 411} \[ \frac{x \sqrt{e+f x^2} \left (3 a d \left (-2 c^2 f^2-5 c d e f+8 d^2 e^2\right )+b c \left (-8 c^2 f^2+c d e f+4 d^2 e^2\right )\right )}{105 c^3 d^2 \left (c+d x^2\right )^{3/2} (d e-c f)}-\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} \left (3 a d \left (c^2 f^2-11 c d e f+8 d^2 e^2\right )+2 b c \left (2 c^2 f^2-c d e f+2 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 c^4 d^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{e+f x^2} \left (6 a d \left (2 c^2 d e f^2+c^3 f^3-12 c d^2 e^2 f+8 d^3 e^3\right )+b c \left (-5 c^2 d e f^2+8 c^3 f^3-5 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{105 c^{7/2} d^{5/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{x \sqrt{e+f x^2} (d e (6 a d+b c)-c f (3 a d+4 b c))}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}-\frac{x \left (e+f x^2\right )^{3/2} (b c-a d)}{7 c d \left (c+d x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 526
Rule 527
Rule 525
Rule 418
Rule 411
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx &=-\frac{(b c-a d) x \left (e+f x^2\right )^{3/2}}{7 c d \left (c+d x^2\right )^{7/2}}-\frac{\int \frac{\sqrt{e+f x^2} \left (-(b c+6 a d) e-(4 b c+3 a d) f x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx}{7 c d}\\ &=\frac{(d (b c+6 a d) e-c (4 b c+3 a d) f) x \sqrt{e+f x^2}}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}-\frac{(b c-a d) x \left (e+f x^2\right )^{3/2}}{7 c d \left (c+d x^2\right )^{7/2}}+\frac{\int \frac{e (4 b c (d e+c f)+3 a d (8 d e+c f))+f (6 a d (3 d e+c f)+b c (3 d e+8 c f)) x^2}{\left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}} \, dx}{35 c^2 d^2}\\ &=\frac{(d (b c+6 a d) e-c (4 b c+3 a d) f) x \sqrt{e+f x^2}}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}+\frac{\left (b c \left (4 d^2 e^2+c d e f-8 c^2 f^2\right )+3 a d \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) x \sqrt{e+f x^2}}{105 c^3 d^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac{(b c-a d) x \left (e+f x^2\right )^{3/2}}{7 c d \left (c+d x^2\right )^{7/2}}-\frac{\int \frac{-e \left (b c \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )+3 a d \left (16 d^2 e^2-16 c d e f-c^2 f^2\right )\right )-f \left (b c \left (4 d^2 e^2+c d e f-8 c^2 f^2\right )+3 a d \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) x^2}{\left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}} \, dx}{105 c^3 d^2 (d e-c f)}\\ &=\frac{(d (b c+6 a d) e-c (4 b c+3 a d) f) x \sqrt{e+f x^2}}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}+\frac{\left (b c \left (4 d^2 e^2+c d e f-8 c^2 f^2\right )+3 a d \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) x \sqrt{e+f x^2}}{105 c^3 d^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac{(b c-a d) x \left (e+f x^2\right )^{3/2}}{7 c d \left (c+d x^2\right )^{7/2}}-\frac{\left (e f \left (3 a d \left (8 d^2 e^2-11 c d e f+c^2 f^2\right )+2 b c \left (2 d^2 e^2-c d e f+2 c^2 f^2\right )\right )\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 c^3 d^2 (d e-c f)^2}+\frac{\left (6 a d \left (8 d^3 e^3-12 c d^2 e^2 f+2 c^2 d e f^2+c^3 f^3\right )+b c \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right ) \int \frac{\sqrt{e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{105 c^3 d^2 (d e-c f)^2}\\ &=\frac{(d (b c+6 a d) e-c (4 b c+3 a d) f) x \sqrt{e+f x^2}}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}+\frac{\left (b c \left (4 d^2 e^2+c d e f-8 c^2 f^2\right )+3 a d \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) x \sqrt{e+f x^2}}{105 c^3 d^2 (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac{(b c-a d) x \left (e+f x^2\right )^{3/2}}{7 c d \left (c+d x^2\right )^{7/2}}+\frac{\left (6 a d \left (8 d^3 e^3-12 c d^2 e^2 f+2 c^2 d e f^2+c^3 f^3\right )+b c \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right ) \sqrt{e+f x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{105 c^{7/2} d^{5/2} (d e-c f)^2 \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{e^{3/2} \sqrt{f} \left (3 a d \left (8 d^2 e^2-11 c d e f+c^2 f^2\right )+2 b c \left (2 d^2 e^2-c d e f+2 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 c^4 d^2 (d e-c f)^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 = 1.83476, size = 545, normalized size = 1.03 \[ \frac{\sqrt{\frac{d}{c}} \left (-x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (-c \left (c+d x^2\right )^2 (d e-c f) \left (3 a d \left (-2 c^2 f^2-5 c d e f+8 d^2 e^2\right )+b c \left (-8 c^2 f^2+c d e f+4 d^2 e^2\right )\right )-\left (c+d x^2\right )^3 \left (6 a d \left (2 c^2 d e f^2+c^3 f^3-12 c d^2 e^2 f+8 d^3 e^3\right )+b c \left (-5 c^2 d e f^2+8 c^3 f^3-5 c d^2 e^2 f+8 d^3 e^3\right )\right )-3 c^2 \left (c+d x^2\right ) (d e-c f)^2 (2 a d (c f+3 d e)+b c (d e-9 c f))+15 c^3 (b c-a d) (d e-c f)^3\right )+i e \left (c+d x^2\right )^3 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (\left (6 a d \left (2 c^2 d e f^2+c^3 f^3-12 c d^2 e^2 f+8 d^3 e^3\right )+b c \left (-5 c^2 d e f^2+8 c^3 f^3-5 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-(c f-d e) \left (3 a d \left (c^2 f^2+16 c d e f-16 d^2 e^2\right )+b c \left (4 c^2 f^2+c d e f-8 d^2 e^2\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )\right )\right )}{105 c^3 d^3 \left (c+d x^2\right )^{7/2} \sqrt{e+f x^2} (d e-c f)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 5113, normalized size = 9.6 \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 (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{9}{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 (\frac{{\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}}{d^{5} x^{10} + 5 \, c d^{4} x^{8} + 10 \, c^{2} d^{3} x^{6} + 10 \, c^{3} d^{2} x^{4} + 5 \, c^{4} d x^{2} + c^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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