Optimal. Leaf size=79 \[ \frac{b x}{a \sqrt{a+b x^2} (b c-a d)}-\frac{d \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} (b c-a d)^{3/2}} \]
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Rubi [A] time = 0.0386197, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {382, 377, 208} \[ \frac{b x}{a \sqrt{a+b x^2} (b c-a d)}-\frac{d \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} (b c-a d)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 382
Rule 377
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )} \, dx &=\frac{b x}{a (b c-a d) \sqrt{a+b x^2}}-\frac{d \int \frac{1}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{b c-a d}\\ &=\frac{b x}{a (b c-a d) \sqrt{a+b x^2}}-\frac{d \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{b c-a d}\\ &=\frac{b x}{a (b c-a d) \sqrt{a+b x^2}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.682355, size = 309, normalized size = 3.91 \[ \frac{x \left (2 d x^2 \left (\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{(b c-a d) x^2}{c \left (b x^2+a\right )}\right )+2 c \left (\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{(b c-a d) x^2}{c \left (b x^2+a\right )}\right )-10 d x^2 \sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}-15 c \sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}+10 d x^2 \tanh ^{-1}\left (\sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )+15 c \tanh ^{-1}\left (\sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )\right )}{5 c^2 \left (a+b x^2\right )^{3/2} \left (\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.013, size = 618, normalized size = 7.8 \begin{align*}{\frac{d}{2\,ad-2\,bc}{\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{2}b+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}}}}}-{\frac{bx}{ \left ( 2\,ad-2\,bc \right ) a}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{2}b+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}}}}}-{\frac{d}{2\,ad-2\,bc}\ln \left ({ \left ( 2\,{\frac{ad-bc}{d}}+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{{\frac{ad-bc}{d}}}\sqrt{ \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) ^{2}b+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}} \right ) \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-{\frac{d}{2\,ad-2\,bc}{\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{2}b-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}}}}}-{\frac{bx}{ \left ( 2\,ad-2\,bc \right ) a}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{2}b-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}}}}}+{\frac{d}{2\,ad-2\,bc}\ln \left ({ \left ( 2\,{\frac{ad-bc}{d}}-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{{\frac{ad-bc}{d}}}\sqrt{ \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) ^{2}b-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}} \right ) \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.36161, size = 911, normalized size = 11.53 \begin{align*} \left [\frac{4 \,{\left (b^{2} c^{2} - a b c d\right )} \sqrt{b x^{2} + a} x -{\left (a b d x^{2} + a^{2} d\right )} \sqrt{b c^{2} - a c d} \log \left (\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \,{\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt{b c^{2} - a c d} \sqrt{b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \,{\left (a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d + a^{4} c d^{2} +{\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )}}, \frac{2 \,{\left (b^{2} c^{2} - a b c d\right )} \sqrt{b x^{2} + a} x +{\left (a b d x^{2} + a^{2} d\right )} \sqrt{-b c^{2} + a c d} \arctan \left (\frac{\sqrt{-b c^{2} + a c d}{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt{b x^{2} + a}}{2 \,{\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} +{\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{2 \,{\left (a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d + a^{4} c d^{2} +{\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )}}\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{1}{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (c + d x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14144, size = 144, normalized size = 1.82 \begin{align*} -\frac{\sqrt{b} d \arctan \left (-\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{\sqrt{-b^{2} c^{2} + a b c d}{\left (b c - a d\right )}} + \frac{b x}{{\left (a b c - a^{2} d\right )} \sqrt{b x^{2} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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