Optimal. Leaf size=296 \[ \frac {12 a^{5/4} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{5 c^{3/2} \sqrt {a+b x^2}}-\frac {24 a^{5/4} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 c^{3/2} \sqrt {a+b x^2}}+\frac {12 b (c x)^{3/2} \sqrt {a+b x^2}}{5 c^3}+\frac {24 a \sqrt {b} \sqrt {c x} \sqrt {a+b x^2}}{5 c^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 \left (a+b x^2\right )^{3/2}}{c \sqrt {c x}} \]
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Rubi [A] time = 0.23, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {277, 279, 329, 305, 220, 1196} \[ \frac {12 a^{5/4} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 c^{3/2} \sqrt {a+b x^2}}-\frac {24 a^{5/4} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 c^{3/2} \sqrt {a+b x^2}}+\frac {12 b (c x)^{3/2} \sqrt {a+b x^2}}{5 c^3}+\frac {24 a \sqrt {b} \sqrt {c x} \sqrt {a+b x^2}}{5 c^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 \left (a+b x^2\right )^{3/2}}{c \sqrt {c x}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 277
Rule 279
Rule 305
Rule 329
Rule 1196
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{3/2}} \, dx &=-\frac {2 \left (a+b x^2\right )^{3/2}}{c \sqrt {c x}}+\frac {(6 b) \int \sqrt {c x} \sqrt {a+b x^2} \, dx}{c^2}\\ &=\frac {12 b (c x)^{3/2} \sqrt {a+b x^2}}{5 c^3}-\frac {2 \left (a+b x^2\right )^{3/2}}{c \sqrt {c x}}+\frac {(12 a b) \int \frac {\sqrt {c x}}{\sqrt {a+b x^2}} \, dx}{5 c^2}\\ &=\frac {12 b (c x)^{3/2} \sqrt {a+b x^2}}{5 c^3}-\frac {2 \left (a+b x^2\right )^{3/2}}{c \sqrt {c x}}+\frac {(24 a b) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{5 c^3}\\ &=\frac {12 b (c x)^{3/2} \sqrt {a+b x^2}}{5 c^3}-\frac {2 \left (a+b x^2\right )^{3/2}}{c \sqrt {c x}}+\frac {\left (24 a^{3/2} \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{5 c^2}-\frac {\left (24 a^{3/2} \sqrt {b}\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^2}\\ &=\frac {12 b (c x)^{3/2} \sqrt {a+b x^2}}{5 c^3}+\frac {24 a \sqrt {b} \sqrt {c x} \sqrt {a+b x^2}}{5 c^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 \left (a+b x^2\right )^{3/2}}{c \sqrt {c x}}-\frac {24 a^{5/4} \sqrt [4]{b} \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^{3/2} \sqrt {a+b x^2}}+\frac {12 a^{5/4} \sqrt [4]{b} \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^{3/2} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 55, normalized size = 0.19 \[ -\frac {2 a x \sqrt {a+b x^2} \, _2F_1\left (-\frac {3}{2},-\frac {1}{4};\frac {3}{4};-\frac {b x^2}{a}\right )}{(c x)^{3/2} \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {c x}}{c^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 208, normalized size = 0.70 \[ \frac {\frac {2 b^{2} x^{4}}{5}-\frac {8 a b \,x^{2}}{5}+\frac {24 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a^{2} \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {12 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a^{2} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{5}-2 a^{2}}{\sqrt {b \,x^{2}+a}\, \sqrt {c x}\, c} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (c\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.48, size = 49, normalized size = 0.17 \[ \frac {a^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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