Optimal. Leaf size=551 \[ \frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a f \left (15 c^2 f^2-11 c d e f+4 d^2 e^2\right )-b e \left (45 c^2 f^2-61 c d e f+24 d^2 e^2\right )\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{105 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (28 a d f (d e-2 c f)-b \left (15 c^2 f^2-43 c d e f+24 d^2 e^2\right )\right )}{105 f^3}+\frac{x \sqrt{c+d x^2} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )-b \left (103 c^2 d e f^2-15 c^3 f^3-128 c d^2 e^2 f+48 d^3 e^3\right )\right )}{105 d f^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )-b \left (103 c^2 d e f^2-15 c^3 f^3-128 c d^2 e^2 f+48 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (-7 a d f-5 b c f+6 b d e)}{35 f^2}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 f} \]
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Rubi [A] time = 0.6285, antiderivative size = 551, 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} \[ -\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} \left (28 a d f (d e-2 c f)-b \left (15 c^2 f^2-43 c d e f+24 d^2 e^2\right )\right )}{105 f^3}+\frac{x \sqrt{c+d x^2} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )-b \left (103 c^2 d e f^2-15 c^3 f^3-128 c d^2 e^2 f+48 d^3 e^3\right )\right )}{105 d f^3 \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a f \left (15 c^2 f^2-11 c d e f+4 d^2 e^2\right )-b e \left (45 c^2 f^2-61 c d e f+24 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )-b \left (103 c^2 d e f^2-15 c^3 f^3-128 c d^2 e^2 f+48 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 d f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (-7 a d f-5 b c f+6 b d e)}{35 f^2}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 f} \]
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 )^{5/2}}{\sqrt{e+f x^2}} \, dx &=\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 f}+\frac{\int \frac{\left (c+d x^2\right )^{3/2} \left (-c (b e-7 a f)+(-6 b d e+5 b c f+7 a d f) x^2\right )}{\sqrt{e+f x^2}} \, dx}{7 f}\\ &=-\frac{(6 b d e-5 b c f-7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 f^2}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 f}+\frac{\int \frac{\sqrt{c+d x^2} \left (-c (7 a f (d e-5 c f)-2 b e (3 d e-5 c f))+\left (-28 a d f (d e-2 c f)+b \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x^2\right )}{\sqrt{e+f x^2}} \, dx}{35 f^2}\\ &=-\frac{\left (28 a d f (d e-2 c f)-b \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 f^3}-\frac{(6 b d e-5 b c f-7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 f^2}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 f}+\frac{\int \frac{c \left (7 a f \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )-b e \left (24 d^2 e^2-61 c d e f+45 c^2 f^2\right )\right )+\left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 f^3}\\ &=-\frac{\left (28 a d f (d e-2 c f)-b \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 f^3}-\frac{(6 b d e-5 b c f-7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 f^2}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 f}+\frac{\left (c \left (7 a f \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )-b e \left (24 d^2 e^2-61 c d e f+45 c^2 f^2\right )\right )\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 f^3}+\frac{\left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{105 f^3}\\ &=\frac{\left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) x \sqrt{c+d x^2}}{105 d f^3 \sqrt{e+f x^2}}-\frac{\left (28 a d f (d e-2 c f)-b \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 f^3}-\frac{(6 b d e-5 b c f-7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 f^2}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 f}+\frac{\sqrt{e} \left (7 a f \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )-b e \left (24 d^2 e^2-61 c d e f+45 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 f^{7/2} \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 (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 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 f^3}\\ &=\frac{\left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) x \sqrt{c+d x^2}}{105 d f^3 \sqrt{e+f x^2}}-\frac{\left (28 a d f (d e-2 c f)-b \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{105 f^3}-\frac{(6 b d e-5 b c f-7 a d f) x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{35 f^2}+\frac{b x \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}}{7 f}-\frac{\sqrt{e} \left (7 a d f \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )-b \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 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 f^{7/2} \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 (7 a f \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right )-b e \left (24 d^2 e^2-61 c d e f+45 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 f^{7/2} \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 = 1.18045, size = 386, normalized size = 0.7 \[ \frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left (4 b e \left (15 c^2 f^2-26 c d e f+12 d^2 e^2\right )-7 a f \left (15 c^2 f^2-19 c d e f+8 d^2 e^2\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )+f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (7 a d f \left (11 c f-4 d e+3 d f x^2\right )+b \left (45 c^2 f^2+c d f \left (45 f x^2-61 e\right )+3 d^2 \left (8 e^2-6 e f x^2+5 f^2 x^4\right )\right )\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (7 a d f \left (23 c^2 f^2-23 c d e f+8 d^2 e^2\right )+b \left (-103 c^2 d e f^2+15 c^3 f^3+128 c d^2 e^2 f-48 d^3 e^3\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{105 f^4 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 1386, 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 \frac{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{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^{2} x^{6} +{\left (2 \, b c d + a d^{2}\right )} x^{4} + a c^{2} +{\left (b c^{2} + 2 \, a c d\right )} x^{2}\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{5}{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{5}{2}}}{\sqrt{f x^{2} + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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