Optimal. Leaf size=410 \[ -\frac{c^{3/2} \sqrt{a+b x^2} \left (a^2 d^2-18 a b c d+b^2 c^2\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{35 b d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 x \sqrt{a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{35 b^2 d \sqrt{c+d x^2}}+\frac{2 \sqrt{c} \sqrt{a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b^2 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{1}{35} x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (-\frac{2 a^2 d}{b}+9 a c+\frac{b c^2}{d}\right )+\frac{d x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 b}+\frac{2 x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (4 b c-a d)}{35 b} \]
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Rubi [A] time = 0.438785, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {416, 528, 531, 418, 492, 411} \[ -\frac{2 x \sqrt{a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{35 b^2 d \sqrt{c+d x^2}}-\frac{c^{3/2} \sqrt{a+b x^2} \left (a^2 d^2-18 a b c d+b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{2 \sqrt{c} \sqrt{a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b^2 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{1}{35} x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (-\frac{2 a^2 d}{b}+9 a c+\frac{b c^2}{d}\right )+\frac{d x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 b}+\frac{2 x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (4 b c-a d)}{35 b} \]
Antiderivative was successfully verified.
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Rule 416
Rule 528
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \, dx &=\frac{d x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 b}+\frac{\int \frac{\left (a+b x^2\right )^{3/2} \left (c (7 b c-a d)+2 d (4 b c-a d) x^2\right )}{\sqrt{c+d x^2}} \, dx}{7 b}\\ &=\frac{2 (4 b c-a d) x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{35 b}+\frac{d x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 b}+\frac{\int \frac{\sqrt{a+b x^2} \left (3 a c d (9 b c-a d)+3 d \left (b^2 c^2+9 a b c d-2 a^2 d^2\right ) x^2\right )}{\sqrt{c+d x^2}} \, dx}{35 b d}\\ &=\frac{1}{35} \left (9 a c+\frac{b c^2}{d}-\frac{2 a^2 d}{b}\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}+\frac{2 (4 b c-a d) x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{35 b}+\frac{d x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 b}+\frac{\int \frac{-3 a c d \left (b^2 c^2-18 a b c d+a^2 d^2\right )-6 d (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{105 b d^2}\\ &=\frac{1}{35} \left (9 a c+\frac{b c^2}{d}-\frac{2 a^2 d}{b}\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}+\frac{2 (4 b c-a d) x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{35 b}+\frac{d x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 b}-\frac{\left (a c \left (b^2 c^2-18 a b c d+a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{35 b d}-\frac{\left (2 (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right )\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{35 b d}\\ &=-\frac{2 (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x \sqrt{a+b x^2}}{35 b^2 d \sqrt{c+d x^2}}+\frac{1}{35} \left (9 a c+\frac{b c^2}{d}-\frac{2 a^2 d}{b}\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}+\frac{2 (4 b c-a d) x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{35 b}+\frac{d x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 b}-\frac{c^{3/2} \left (b^2 c^2-18 a b c d+a^2 d^2\right ) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b d^{3/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{\left (2 c (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right )\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{35 b^2 d}\\ &=-\frac{2 (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x \sqrt{a+b x^2}}{35 b^2 d \sqrt{c+d x^2}}+\frac{1}{35} \left (9 a c+\frac{b c^2}{d}-\frac{2 a^2 d}{b}\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}+\frac{2 (4 b c-a d) x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{35 b}+\frac{d x \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{7 b}+\frac{2 \sqrt{c} (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b^2 d^{3/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}-\frac{c^{3/2} \left (b^2 c^2-18 a b c d+a^2 d^2\right ) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{35 b d^{3/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.583425, size = 302, normalized size = 0.74 \[ \frac{-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (8 a^2 b c d^2+a^3 d^3-11 a b^2 c^2 d+2 b^3 c^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{b}{a}}\right ),\frac{a d}{b c}\right )+d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a^2 d^2+a b d \left (17 c+8 d x^2\right )+b^2 \left (c^2+8 c d x^2+5 d^2 x^4\right )\right )+2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (-5 a^2 b c d^2+a^3 d^3-5 a b^2 c^2 d+b^3 c^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{35 b d^2 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 780, normalized size = 1.9 \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{\left (b x^{2} + a\right )}^{\frac{3}{2}}{\left (d x^{2} + c\right )}^{\frac{3}{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 ({\left (b d x^{4} +{\left (b c + a d\right )} x^{2} + a c\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}, 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 \left (a + b x^{2}\right )^{\frac{3}{2}} \left (c + d x^{2}\right )^{\frac{3}{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{\left (b x^{2} + a\right )}^{\frac{3}{2}}{\left (d x^{2} + c\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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