Optimal. Leaf size=297 \[ \frac {12 \sqrt [4]{a} b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{5 c^{7/2} \sqrt {a+b x^2}}-\frac {24 \sqrt [4]{a} b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 c^{7/2} \sqrt {a+b x^2}}+\frac {24 b^{3/2} \sqrt {c x} \sqrt {a+b x^2}}{5 c^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {12 b \sqrt {a+b x^2}}{5 c^3 \sqrt {c x}}-\frac {2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}} \]
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Rubi [A] time = 0.23, 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 = {277, 329, 305, 220, 1196} \[ \frac {24 b^{3/2} \sqrt {c x} \sqrt {a+b x^2}}{5 c^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {12 \sqrt [4]{a} b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 c^{7/2} \sqrt {a+b x^2}}-\frac {24 \sqrt [4]{a} b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 c^{7/2} \sqrt {a+b x^2}}-\frac {12 b \sqrt {a+b x^2}}{5 c^3 \sqrt {c x}}-\frac {2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 277
Rule 305
Rule 329
Rule 1196
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{7/2}} \, dx &=-\frac {2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}}+\frac {(6 b) \int \frac {\sqrt {a+b x^2}}{(c x)^{3/2}} \, dx}{5 c^2}\\ &=-\frac {12 b \sqrt {a+b x^2}}{5 c^3 \sqrt {c x}}-\frac {2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}}+\frac {\left (12 b^2\right ) \int \frac {\sqrt {c x}}{\sqrt {a+b x^2}} \, dx}{5 c^4}\\ &=-\frac {12 b \sqrt {a+b x^2}}{5 c^3 \sqrt {c x}}-\frac {2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}}+\frac {\left (24 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{5 c^5}\\ &=-\frac {12 b \sqrt {a+b x^2}}{5 c^3 \sqrt {c x}}-\frac {2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}}+\frac {\left (24 \sqrt {a} b^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{5 c^4}-\frac {\left (24 \sqrt {a} b^{3/2}\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 )}{5 c^4}\\ &=-\frac {12 b \sqrt {a+b x^2}}{5 c^3 \sqrt {c x}}+\frac {24 b^{3/2} \sqrt {c x} \sqrt {a+b x^2}}{5 c^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}}-\frac {24 \sqrt [4]{a} b^{5/4} \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 )}{5 c^{7/2} \sqrt {a+b x^2}}+\frac {12 \sqrt [4]{a} b^{5/4} \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 )}{5 c^{7/2} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 57, normalized size = 0.19 \[ -\frac {2 a x \sqrt {a+b x^2} \, _2F_1\left (-\frac {3}{2},-\frac {5}{4};-\frac {1}{4};-\frac {b x^2}{a}\right )}{5 (c x)^{7/2} \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {c x}}{c^{4} x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 216, normalized size = 0.73 \[ \frac {-\frac {14 b^{2} x^{4}}{5}+\frac {24 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a b \,x^{2} \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {12 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a b \,x^{2} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {16 a b \,x^{2}}{5}-\frac {2 a^{2}}{5}}{\sqrt {b \,x^{2}+a}\, \sqrt {c x}\, c^{3} x^{2}} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (c\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 10.93, size = 53, normalized size = 0.18 \[ \frac {a^{\frac {3}{2}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {5}{4} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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