Optimal. Leaf size=331 \[ \frac {4 b^{9/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{15 a^{3/4} c^{11/2} \sqrt {a+b x^2}}-\frac {8 b^{9/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 a^{3/4} c^{11/2} \sqrt {a+b x^2}}+\frac {8 b^{5/2} \sqrt {c x} \sqrt {a+b x^2}}{15 a c^6 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {8 b^2 \sqrt {a+b x^2}}{15 a c^5 \sqrt {c x}}-\frac {4 b \sqrt {a+b x^2}}{15 c^3 (c x)^{5/2}}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}} \]
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Rubi [A] time = 0.27, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {277, 325, 329, 305, 220, 1196} \[ \frac {4 b^{9/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 a^{3/4} c^{11/2} \sqrt {a+b x^2}}-\frac {8 b^{9/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 a^{3/4} c^{11/2} \sqrt {a+b x^2}}+\frac {8 b^{5/2} \sqrt {c x} \sqrt {a+b x^2}}{15 a c^6 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {8 b^2 \sqrt {a+b x^2}}{15 a c^5 \sqrt {c x}}-\frac {4 b \sqrt {a+b x^2}}{15 c^3 (c x)^{5/2}}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 277
Rule 305
Rule 325
Rule 329
Rule 1196
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{11/2}} \, dx &=-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}+\frac {(2 b) \int \frac {\sqrt {a+b x^2}}{(c x)^{7/2}} \, dx}{3 c^2}\\ &=-\frac {4 b \sqrt {a+b x^2}}{15 c^3 (c x)^{5/2}}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}+\frac {\left (4 b^2\right ) \int \frac {1}{(c x)^{3/2} \sqrt {a+b x^2}} \, dx}{15 c^4}\\ &=-\frac {4 b \sqrt {a+b x^2}}{15 c^3 (c x)^{5/2}}-\frac {8 b^2 \sqrt {a+b x^2}}{15 a c^5 \sqrt {c x}}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}+\frac {\left (4 b^3\right ) \int \frac {\sqrt {c x}}{\sqrt {a+b x^2}} \, dx}{15 a c^6}\\ &=-\frac {4 b \sqrt {a+b x^2}}{15 c^3 (c x)^{5/2}}-\frac {8 b^2 \sqrt {a+b x^2}}{15 a c^5 \sqrt {c x}}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}+\frac {\left (8 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{15 a c^7}\\ &=-\frac {4 b \sqrt {a+b x^2}}{15 c^3 (c x)^{5/2}}-\frac {8 b^2 \sqrt {a+b x^2}}{15 a c^5 \sqrt {c x}}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}+\frac {\left (8 b^{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 {a} c^6}-\frac {\left (8 b^{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 {a} c^6}\\ &=-\frac {4 b \sqrt {a+b x^2}}{15 c^3 (c x)^{5/2}}-\frac {8 b^2 \sqrt {a+b x^2}}{15 a c^5 \sqrt {c x}}+\frac {8 b^{5/2} \sqrt {c x} \sqrt {a+b x^2}}{15 a c^6 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}-\frac {8 b^{9/4} \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 a^{3/4} c^{11/2} \sqrt {a+b x^2}}+\frac {4 b^{9/4} \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 a^{3/4} c^{11/2} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 57, normalized size = 0.17 \[ -\frac {2 a x \sqrt {a+b x^2} \, _2F_1\left (-\frac {9}{4},-\frac {3}{2};-\frac {5}{4};-\frac {b x^2}{a}\right )}{9 (c x)^{11/2} \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {c x}}{c^{6} x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 234, normalized size = 0.71 \[ \frac {-\frac {8 b^{3} x^{6}}{15}+\frac {8 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a \,b^{2} x^{4} \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{15}-\frac {4 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a \,b^{2} x^{4} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{15}-\frac {46 a \,b^{2} x^{4}}{45}-\frac {32 a^{2} b \,x^{2}}{45}-\frac {2 a^{3}}{9}}{\sqrt {b \,x^{2}+a}\, \sqrt {c x}\, a \,c^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (c\,x\right )}^{11/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 83.79, size = 53, normalized size = 0.16 \[ \frac {a^{\frac {3}{2}} \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {3}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac {11}{2}} x^{\frac {9}{2}} \Gamma \left (- \frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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