Optimal. Leaf size=297 \[ \frac {4 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}-\frac {8 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}+\frac {8 a^2 \sqrt {c x} \sqrt {a+b x^2}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {279, 329, 305, 220, 1196} \[ \frac {4 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}-\frac {8 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}+\frac {8 a^2 \sqrt {c x} \sqrt {a+b x^2}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 279
Rule 305
Rule 329
Rule 1196
Rubi steps
\begin {align*} \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx &=\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac {1}{3} (2 a) \int \sqrt {c x} \sqrt {a+b x^2} \, dx\\ &=\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac {1}{15} \left (4 a^2\right ) \int \frac {\sqrt {c x}}{\sqrt {a+b x^2}} \, dx\\ &=\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac {\left (8 a^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{15 c}\\ &=\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac {\left (8 a^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{15 \sqrt {b}}-\frac {\left (8 a^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a} c}}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{15 \sqrt {b}}\\ &=\frac {4 a (c x)^{3/2} \sqrt {a+b x^2}}{15 c}+\frac {8 a^2 \sqrt {c x} \sqrt {a+b x^2}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}-\frac {8 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}+\frac {4 a^{9/4} \sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 57, normalized size = 0.19 \[ \frac {2 a x \sqrt {c x} \sqrt {a+b x^2} \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^2}{a}\right )}{3 \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {c x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 218, normalized size = 0.73 \[ \frac {2 \sqrt {c x}\, \left (5 b^{3} x^{6}+16 a \,b^{2} x^{4}+11 a^{2} b \,x^{2}+12 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a^{3} \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-6 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a^{3} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )\right )}{45 \sqrt {b \,x^{2}+a}\, b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {c\,x}\,{\left (b\,x^2+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 3.40, size = 46, normalized size = 0.15 \[ \frac {a^{\frac {3}{2}} \sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________