Optimal. Leaf size=383 \[ -\frac{e \left (c+d x^2\right ) (7 b c-8 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3 x}+\frac{e \left (c+d x^2\right ) (3 b c-4 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{3 c^2 d x^3}+\frac{d e x (7 b c-8 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3}+\frac{b e (3 b c-4 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a c^{3/2} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{d} e (7 b c-8 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{5/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3} \]
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Rubi [A] time = 0.631784, antiderivative size = 383, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {6719, 468, 583, 531, 418, 492, 411} \[ -\frac{e \left (c+d x^2\right ) (7 b c-8 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3 x}+\frac{e \left (c+d x^2\right ) (3 b c-4 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{3 c^2 d x^3}+\frac{d e x (7 b c-8 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3}+\frac{b e (3 b c-4 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a c^{3/2} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{d} e (7 b c-8 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{5/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3} \]
Antiderivative was successfully verified.
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Rule 6719
Rule 468
Rule 583
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^4} \, dx &=\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{\left (a+b x^2\right )^{3/2}}{x^4 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}-\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{a (3 b c-4 a d)+b (2 b c-3 a d) x^2}{x^4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{c d \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}+\frac{(3 b c-4 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^2 d x^3}+\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{a^2 d (7 b c-8 a d)+a b d (3 b c-4 a d) x^2}{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 a c^2 d \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}+\frac{(3 b c-4 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^2 d x^3}-\frac{(7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^3 x}-\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{-a^2 b c d (3 b c-4 a d)-a^2 b d^2 (7 b c-8 a d) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 a^2 c^3 d \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}+\frac{(3 b c-4 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^2 d x^3}-\frac{(7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^3 x}+\frac{\left (b d (7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 c^3 \sqrt{a+b x^2}}+\frac{\left (b (3 b c-4 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 c^2 \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}+\frac{d (7 b c-8 a d) e x \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3}+\frac{(3 b c-4 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^2 d x^3}-\frac{(7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^3 x}+\frac{b (3 b c-4 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a c^{3/2} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\left (d (7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c^2 \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}+\frac{d (7 b c-8 a d) e x \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3}+\frac{(3 b c-4 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^2 d x^3}-\frac{(7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^3 x}-\frac{\sqrt{d} (7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{5/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b (3 b c-4 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a c^{3/2} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\\ \end{align*}
Mathematica [C] time = 0.454519, size = 275, normalized size = 0.72 \[ \frac{e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (-\sqrt{\frac{b}{a}} \left (a^2 \left (c^2-4 c d x^2-8 d^2 x^4\right )+a b x^2 \left (5 c^2+3 c d x^2-8 d^2 x^4\right )+b^2 c x^4 \left (4 c+7 d x^2\right )\right )-4 i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (8 a d-7 b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )\right )}{3 c^3 x^3 \sqrt{\frac{b}{a}} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 791, normalized size = 2.1 \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 (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}}}{x^{4}}\,{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 e x^{2} + a e\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{d x^{6} + c x^{4}}, 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 (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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