Optimal. Leaf size=329 \[ -\frac {4 a^{13/4} c^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a+b x^2}}+\frac {8 a^{13/4} c^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a+b x^2}}-\frac {8 a^3 c^2 \sqrt {c x} \sqrt {a+b x^2}}{65 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {8 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}{195 b}+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac {4 a (c x)^{7/2} \sqrt {a+b x^2}}{39 c} \]
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Rubi [A] time = 0.26, antiderivative size = 329, 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 = {279, 321, 329, 305, 220, 1196} \[ -\frac {8 a^3 c^2 \sqrt {c x} \sqrt {a+b x^2}}{65 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {4 a^{13/4} c^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a+b x^2}}+\frac {8 a^{13/4} c^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a+b x^2}}+\frac {8 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}{195 b}+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac {4 a (c x)^{7/2} \sqrt {a+b x^2}}{39 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 279
Rule 305
Rule 321
Rule 329
Rule 1196
Rubi steps
\begin {align*} \int (c x)^{5/2} \left (a+b x^2\right )^{3/2} \, dx &=\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac {1}{13} (6 a) \int (c x)^{5/2} \sqrt {a+b x^2} \, dx\\ &=\frac {4 a (c x)^{7/2} \sqrt {a+b x^2}}{39 c}+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac {1}{39} \left (4 a^2\right ) \int \frac {(c x)^{5/2}}{\sqrt {a+b x^2}} \, dx\\ &=\frac {8 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}{195 b}+\frac {4 a (c x)^{7/2} \sqrt {a+b x^2}}{39 c}+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}-\frac {\left (4 a^3 c^2\right ) \int \frac {\sqrt {c x}}{\sqrt {a+b x^2}} \, dx}{65 b}\\ &=\frac {8 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}{195 b}+\frac {4 a (c x)^{7/2} \sqrt {a+b x^2}}{39 c}+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}-\frac {\left (8 a^3 c\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{65 b}\\ &=\frac {8 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}{195 b}+\frac {4 a (c x)^{7/2} \sqrt {a+b x^2}}{39 c}+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}-\frac {\left (8 a^{7/2} c^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{65 b^{3/2}}+\frac {\left (8 a^{7/2} c^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 )}{65 b^{3/2}}\\ &=\frac {8 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}{195 b}+\frac {4 a (c x)^{7/2} \sqrt {a+b x^2}}{39 c}-\frac {8 a^3 c^2 \sqrt {c x} \sqrt {a+b x^2}}{65 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac {8 a^{13/4} c^{5/2} \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 )}{65 b^{7/4} \sqrt {a+b x^2}}-\frac {4 a^{13/4} c^{5/2} \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 )}{65 b^{7/4} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 89, normalized size = 0.27 \[ \frac {2 c (c x)^{3/2} \sqrt {a+b x^2} \left (\left (a+b x^2\right )^2 \sqrt {\frac {b x^2}{a}+1}-a^2 \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^2}{a}\right )\right )}{13 b \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b c^{2} x^{4} + a c^{2} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {c x}, x\right ) \]
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 \[ \int {\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 232, normalized size = 0.71 \[ -\frac {2 \sqrt {c x}\, \left (-15 b^{4} x^{8}-40 a \,b^{3} x^{6}-29 a^{2} b^{2} x^{4}-4 a^{3} b \,x^{2}+12 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a^{4} \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-6 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a^{4} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )\right ) c^{2}}{195 \sqrt {b \,x^{2}+a}\, b^{2} x} \]
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 \[ \int {\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (c\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 18.19, size = 46, normalized size = 0.14 \[ \frac {a^{\frac {3}{2}} c^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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