Optimal. Leaf size=336 \[ \frac{c^{3/2} \sqrt{a+b x^2} (9 b c-a d) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{15 b \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right )}{15 b^2 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{d x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 b}+\frac{2 x \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-a d)}{15 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.280561, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{x \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right )}{15 b^2 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{c^{3/2} \sqrt{a+b x^2} (9 b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{d x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 b}+\frac{2 x \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-a d)}{15 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 416
Rule 528
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \sqrt{a+b x^2} \left (c+d x^2\right )^{3/2} \, dx &=\frac{d x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 b}+\frac{\int \frac{\sqrt{a+b x^2} \left (c (5 b c-a d)+2 d (3 b c-a d) x^2\right )}{\sqrt{c+d x^2}} \, dx}{5 b}\\ &=\frac{2 (3 b c-a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 b}+\frac{d x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 b}+\frac{\int \frac{a c d (9 b c-a d)+d \left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{15 b d}\\ &=\frac{2 (3 b c-a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 b}+\frac{d x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 b}+\frac{(a c (9 b c-a d)) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{15 b}+\frac{\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{15 b}\\ &=\frac{\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) x \sqrt{a+b x^2}}{15 b^2 \sqrt{c+d x^2}}+\frac{2 (3 b c-a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 b}+\frac{d x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 b}+\frac{c^{3/2} (9 b c-a d) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}-\frac{\left (c \left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right )\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 b^2}\\ &=\frac{\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) x \sqrt{a+b x^2}}{15 b^2 \sqrt{c+d x^2}}+\frac{2 (3 b c-a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 b}+\frac{d x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 b}-\frac{\sqrt{c} \left (3 b^2 c^2+7 a b c d-2 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 )}{15 b^2 \sqrt{d} \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} (9 b c-a d) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b \sqrt{d} \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 = 0.410342, size = 246, normalized size = 0.73 \[ \frac{-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (a^2 d^2+2 a b c d-3 b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{b}{a}}\right ),\frac{a d}{b c}\right )+i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (2 a^2 d^2-7 a b c d-3 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a d+6 b c+3 b d x^2\right )}{15 b d \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 545, normalized size = 1.6 \begin{align*}{\frac{1}{15\,d \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) b}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 3\,\sqrt{-{\frac{b}{a}}}{x}^{7}{b}^{2}{d}^{3}+4\,\sqrt{-{\frac{b}{a}}}{x}^{5}ab{d}^{3}+9\,\sqrt{-{\frac{b}{a}}}{x}^{5}{b}^{2}c{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{3}{a}^{2}{d}^{3}+10\,\sqrt{-{\frac{b}{a}}}{x}^{3}abc{d}^{2}+6\,\sqrt{-{\frac{b}{a}}}{x}^{3}{b}^{2}{c}^{2}d+\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}c{d}^{2}+2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}d-3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{2}{c}^{3}-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}c{d}^{2}+7\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}d+3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{2}{c}^{3}+\sqrt{-{\frac{b}{a}}}x{a}^{2}c{d}^{2}+6\,\sqrt{-{\frac{b}{a}}}xab{c}^{2}d \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b x^{2} + a}{\left (d x^{2} + c\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b x^{2} + a}{\left (d x^{2} + c\right )}^{\frac{3}{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b x^{2}} \left (c + d x^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b x^{2} + a}{\left (d x^{2} + c\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]