Optimal. Leaf size=204 \[ \frac{2 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-11 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{b x^2+c x^4} (5 b B-11 A c)}{231 c^2 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x^2+c x^4} (5 b B-11 A c)}{77 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.29968, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2039, 2021, 2024, 2032, 329, 220} \[ \frac{2 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-11 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{b x^2+c x^4} (5 b B-11 A c)}{231 c^2 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x^2+c x^4} (5 b B-11 A c)}{77 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2039
Rule 2021
Rule 2024
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \sqrt{x} \left (A+B x^2\right ) \sqrt{b x^2+c x^4} \, dx &=\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}}-\frac{\left (2 \left (\frac{5 b B}{2}-\frac{11 A c}{2}\right )\right ) \int \sqrt{x} \sqrt{b x^2+c x^4} \, dx}{11 c}\\ &=-\frac{2 (5 b B-11 A c) x^{3/2} \sqrt{b x^2+c x^4}}{77 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}}-\frac{(2 b (5 b B-11 A c)) \int \frac{x^{5/2}}{\sqrt{b x^2+c x^4}} \, dx}{77 c}\\ &=-\frac{4 b (5 b B-11 A c) \sqrt{b x^2+c x^4}}{231 c^2 \sqrt{x}}-\frac{2 (5 b B-11 A c) x^{3/2} \sqrt{b x^2+c x^4}}{77 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}}+\frac{\left (2 b^2 (5 b B-11 A c)\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{231 c^2}\\ &=-\frac{4 b (5 b B-11 A c) \sqrt{b x^2+c x^4}}{231 c^2 \sqrt{x}}-\frac{2 (5 b B-11 A c) x^{3/2} \sqrt{b x^2+c x^4}}{77 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}}+\frac{\left (2 b^2 (5 b B-11 A c) x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{231 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{4 b (5 b B-11 A c) \sqrt{b x^2+c x^4}}{231 c^2 \sqrt{x}}-\frac{2 (5 b B-11 A c) x^{3/2} \sqrt{b x^2+c x^4}}{77 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}}+\frac{\left (4 b^2 (5 b B-11 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 )}{231 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{4 b (5 b B-11 A c) \sqrt{b x^2+c x^4}}{231 c^2 \sqrt{x}}-\frac{2 (5 b B-11 A c) x^{3/2} \sqrt{b x^2+c x^4}}{77 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}}+\frac{2 b^{7/4} (5 b B-11 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 )}{231 c^{9/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.129684, size = 111, normalized size = 0.54 \[ \frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (b (5 b B-11 A c) \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right )-\left (b+c x^2\right ) \sqrt{\frac{c x^2}{b}+1} \left (-11 A c+5 b B-7 B c x^2\right )\right )}{77 c^2 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 283, normalized size = 1.4 \begin{align*} -{\frac{2}{ \left ( 231\,c{x}^{2}+231\,b \right ){c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( -21\,B{x}^{7}{c}^{4}+11\,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}{b}^{2}c-33\,A{x}^{5}{c}^{4}-5\,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}{b}^{3}-27\,B{x}^{5}b{c}^{3}-55\,A{x}^{3}b{c}^{3}+4\,B{x}^{3}{b}^{2}{c}^{2}-22\,Ax{b}^{2}{c}^{2}+10\,Bx{b}^{3}c \right ){x}^{-{\frac{3}{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 \sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )} \sqrt{x}\,{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{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )} \sqrt{x}, 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{x} \sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, 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{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )} \sqrt{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]