Optimal. Leaf size=639 \[ \frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (3 b c (3 d e-2 c f)-a d (4 d e-c f)) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{15 c^3 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{b^2 e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{e+f x^2} \left (a d \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )-3 b c \left (c^2 f^2-6 c d e f+6 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{x \sqrt{e+f x^2} (3 b c (3 d e-c f)-2 a d (c f+2 d e))}{15 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac{x \sqrt{e+f x^2} (d e-c f)}{5 c \left (c+d x^2\right )^{5/2} (b c-a d)}-\frac{b \sqrt{d} \sqrt{e+f x^2} (b e-a f) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{\sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.779407, antiderivative size = 639, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {544, 541, 539, 411, 526, 527, 525, 418} \[ \frac{b^2 e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{e+f x^2} \left (a d \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )-3 b c \left (c^2 f^2-6 c d e f+6 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (3 b c (3 d e-2 c f)-a d (4 d e-c f)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (3 b c (3 d e-c f)-2 a d (c f+2 d e))}{15 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac{x \sqrt{e+f x^2} (d e-c f)}{5 c \left (c+d x^2\right )^{5/2} (b c-a d)}-\frac{b \sqrt{d} \sqrt{e+f x^2} (b e-a f) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{\sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 544
Rule 541
Rule 539
Rule 411
Rule 526
Rule 527
Rule 525
Rule 418
Rubi steps
\begin{align*} \int \frac{\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx &=-\frac{\int \frac{\sqrt{e+f x^2} \left (2 b c d e-a d^2 e-b c^2 f+d^2 (b e-a f) x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx}{(b c-a d)^2}+\frac{(b (b e-a f)) \int \frac{\sqrt{e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{(b c-a d)^2}\\ &=-\frac{(d e-c f) x \sqrt{e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac{\int \frac{-d e (b c (9 d e-4 c f)-a d (4 d e+c f))-d f (b c (8 d e-3 c f)-a d (3 d e+2 c f)) x^2}{\left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}} \, dx}{5 c d (b c-a d)^2}+\frac{\left (b^2 (b e-a f)\right ) \int \frac{\sqrt{e+f x^2}}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{(b c-a d)^3}-\frac{(b d (b e-a f)) \int \frac{\sqrt{e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{(b c-a d)^3}\\ &=-\frac{(d e-c f) x \sqrt{e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac{(3 b c (3 d e-c f)-2 a d (2 d e+c f)) x \sqrt{e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac{b \sqrt{d} (b e-a f) \sqrt{e+f x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{\sqrt{c} (b c-a d)^3 \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{b^2 e^{3/2} (b e-a f) \sqrt{c+d x^2} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a c (b c-a d)^3 \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}-\frac{\int \frac{d e (d e-c f) (9 b c (2 d e-c f)-a d (8 d e+c f))+d f (d e-c f) (3 b c (3 d e-c f)-2 a d (2 d e+c f)) x^2}{\left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}} \, dx}{15 c^2 d (b c-a d)^2 (d e-c f)}\\ &=-\frac{(d e-c f) x \sqrt{e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac{(3 b c (3 d e-c f)-2 a d (2 d e+c f)) x \sqrt{e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac{b \sqrt{d} (b e-a f) \sqrt{e+f x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{\sqrt{c} (b c-a d)^3 \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{b^2 e^{3/2} (b e-a f) \sqrt{c+d x^2} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a c (b c-a d)^3 \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}-\frac{\left (e f \left (a d (4 d e-c f)-b \left (9 c d e-6 c^2 f\right )\right )\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 c^2 (b c-a d)^2 (d e-c f)}+\frac{\left (a d \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-3 b c \left (6 d^2 e^2-6 c d e f+c^2 f^2\right )\right ) \int \frac{\sqrt{e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 c^2 (b c-a d)^2 (d e-c f)}\\ &=-\frac{(d e-c f) x \sqrt{e+f x^2}}{5 c (b c-a d) \left (c+d x^2\right )^{5/2}}-\frac{(3 b c (3 d e-c f)-2 a d (2 d e+c f)) x \sqrt{e+f x^2}}{15 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac{b \sqrt{d} (b e-a f) \sqrt{e+f x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{\sqrt{c} (b c-a d)^3 \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{\left (a d \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )-3 b c \left (6 d^2 e^2-6 c d e f+c^2 f^2\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 )}{15 c^{5/2} \sqrt{d} (b c-a d)^2 (d e-c f) \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 (a d (4 d e-c f)-b \left (9 c d e-6 c^2 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 c^3 (b c-a d)^2 (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{b^2 e^{3/2} (b e-a f) \sqrt{c+d x^2} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a c (b c-a d)^3 \sqrt{f} \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 = 2.89116, size = 570, normalized size = 0.89 \[ \frac{-a x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (\left (c+d x^2\right )^2 \left (a^2 d^2 \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )+2 a b c d \left (7 c^2 f^2+3 c d e f-13 d^2 e^2\right )+3 b^2 c^2 \left (c^2 f^2-11 c d e f+11 d^2 e^2\right )\right )+3 c^2 (b c-a d)^2 (d e-c f)^2+c \left (c+d x^2\right ) (b c-a d) (c f-d e) (2 a d (c f+2 d e)+3 b c (c f-3 d e))\right )+i \left (c+d x^2\right )^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left ((d e-c f) \left (a \left (a^2 d^2 e (c f+8 d e)+a b c \left (15 c^2 f^2-7 c d e f-26 d^2 e^2\right )+3 b^2 c^2 e (11 d e-8 c f)\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )-15 b c^3 (b e-a f)^2 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )+a e \left (a^2 d^2 \left (2 c^2 f^2+3 c d e f-8 d^2 e^2\right )-2 a b c d \left (7 c^2 f^2+3 c d e f-13 d^2 e^2\right )-3 b^2 c^2 \left (c^2 f^2-11 c d e f+11 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{15 a c^3 \sqrt{\frac{d}{c}} \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2} (b c-a d)^3 (d e-c f)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 6211, normalized size = 9.7 \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 (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{7}{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(-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 (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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