Optimal. Leaf size=200 \[ \frac{4 b^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 A c+3 b B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{21 \sqrt [4]{c} \sqrt{b x^2+c x^4}}+\frac{2 \left (b x^2+c x^4\right )^{3/2} (7 A c+3 b B)}{21 b x^{5/2}}+\frac{4 \sqrt{b x^2+c x^4} (7 A c+3 b B)}{21 \sqrt{x}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.3222, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2038, 2021, 2032, 329, 220} \[ \frac{4 b^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 A c+3 b B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 \sqrt [4]{c} \sqrt{b x^2+c x^4}}+\frac{2 \left (b x^2+c x^4\right )^{3/2} (7 A c+3 b B)}{21 b x^{5/2}}+\frac{4 \sqrt{b x^2+c x^4} (7 A c+3 b B)}{21 \sqrt{x}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2038
Rule 2021
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{11/2}} \, dx &=-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}-\frac{\left (2 \left (-\frac{3 b B}{2}-\frac{7 A c}{2}\right )\right ) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{7/2}} \, dx}{3 b}\\ &=\frac{2 (3 b B+7 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b x^{5/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}+\frac{1}{7} (2 (3 b B+7 A c)) \int \frac{\sqrt{b x^2+c x^4}}{x^{3/2}} \, dx\\ &=\frac{4 (3 b B+7 A c) \sqrt{b x^2+c x^4}}{21 \sqrt{x}}+\frac{2 (3 b B+7 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b x^{5/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}+\frac{1}{21} (4 b (3 b B+7 A c)) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{4 (3 b B+7 A c) \sqrt{b x^2+c x^4}}{21 \sqrt{x}}+\frac{2 (3 b B+7 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b x^{5/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}+\frac{\left (4 b (3 b B+7 A c) x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{21 \sqrt{b x^2+c x^4}}\\ &=\frac{4 (3 b B+7 A c) \sqrt{b x^2+c x^4}}{21 \sqrt{x}}+\frac{2 (3 b B+7 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b x^{5/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}+\frac{\left (8 b (3 b B+7 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{21 \sqrt{b x^2+c x^4}}\\ &=\frac{4 (3 b B+7 A c) \sqrt{b x^2+c x^4}}{21 \sqrt{x}}+\frac{2 (3 b B+7 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b x^{5/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}+\frac{4 b^{3/4} (3 b B+7 A c) x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 \sqrt [4]{c} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0548848, size = 101, normalized size = 0.5 \[ -\frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (A \left (b+c x^2\right )^2 \sqrt{\frac{c x^2}{b}+1}-b x^2 (7 A c+3 b B) \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right )\right )}{3 b x^{5/2} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.015, size = 260, normalized size = 1.3 \begin{align*}{\frac{2}{21\, \left ( c{x}^{2}+b \right ) ^{2}c} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 14\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}xbc+6\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}x{b}^{2}+3\,B{c}^{3}{x}^{6}+7\,A{x}^{4}{c}^{3}+12\,B{x}^{4}b{c}^{2}+9\,B{x}^{2}{b}^{2}c-7\,A{b}^{2}c \right ){x}^{-{\frac{9}{2}}}} \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 \frac{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{x^{\frac{11}{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 (\frac{{\left (B c x^{4} +{\left (B b + A c\right )} x^{2} + A b\right )} \sqrt{c x^{4} + b x^{2}}}{x^{\frac{7}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{x^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]