Optimal. Leaf size=212 \[ \frac {4 a^{15/4} c^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a+b x^2}}-\frac {8 a^3 c^3 \sqrt {c x} \sqrt {a+b x^2}}{231 b^2}+\frac {8 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}{385 b}+\frac {2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac {4 a (c x)^{9/2} \sqrt {a+b x^2}}{55 c} \]
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Rubi [A] time = 0.13, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {279, 321, 329, 220} \[ -\frac {8 a^3 c^3 \sqrt {c x} \sqrt {a+b x^2}}{231 b^2}+\frac {4 a^{15/4} c^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a+b x^2}}+\frac {8 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}{385 b}+\frac {2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac {4 a (c x)^{9/2} \sqrt {a+b x^2}}{55 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 279
Rule 321
Rule 329
Rubi steps
\begin {align*} \int (c x)^{7/2} \left (a+b x^2\right )^{3/2} \, dx &=\frac {2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac {1}{5} (2 a) \int (c x)^{7/2} \sqrt {a+b x^2} \, dx\\ &=\frac {4 a (c x)^{9/2} \sqrt {a+b x^2}}{55 c}+\frac {2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac {1}{55} \left (4 a^2\right ) \int \frac {(c x)^{7/2}}{\sqrt {a+b x^2}} \, dx\\ &=\frac {8 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}{385 b}+\frac {4 a (c x)^{9/2} \sqrt {a+b x^2}}{55 c}+\frac {2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}-\frac {\left (4 a^3 c^2\right ) \int \frac {(c x)^{3/2}}{\sqrt {a+b x^2}} \, dx}{77 b}\\ &=-\frac {8 a^3 c^3 \sqrt {c x} \sqrt {a+b x^2}}{231 b^2}+\frac {8 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}{385 b}+\frac {4 a (c x)^{9/2} \sqrt {a+b x^2}}{55 c}+\frac {2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac {\left (4 a^4 c^4\right ) \int \frac {1}{\sqrt {c x} \sqrt {a+b x^2}} \, dx}{231 b^2}\\ &=-\frac {8 a^3 c^3 \sqrt {c x} \sqrt {a+b x^2}}{231 b^2}+\frac {8 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}{385 b}+\frac {4 a (c x)^{9/2} \sqrt {a+b x^2}}{55 c}+\frac {2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac {\left (8 a^4 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{231 b^2}\\ &=-\frac {8 a^3 c^3 \sqrt {c x} \sqrt {a+b x^2}}{231 b^2}+\frac {8 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}{385 b}+\frac {4 a (c x)^{9/2} \sqrt {a+b x^2}}{55 c}+\frac {2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac {4 a^{15/4} c^{7/2} \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 )}{231 b^{9/4} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 102, normalized size = 0.48 \[ \frac {2 c^3 \sqrt {c x} \sqrt {a+b x^2} \left (5 a^3 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )-\left (5 a-11 b x^2\right ) \left (a+b x^2\right )^2 \sqrt {\frac {b x^2}{a}+1}\right )}{165 b^2 \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b c^{3} x^{5} + a c^{3} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {c x}, x\right ) \]
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 \[ \int {\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 163, normalized size = 0.77 \[ \frac {2 \sqrt {c x}\, \left (77 b^{5} x^{9}+196 a \,b^{4} x^{7}+131 a^{2} b^{3} x^{5}-8 a^{3} b^{2} x^{3}-20 a^{4} b x +10 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, a^{4} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )\right ) c^{3}}{1155 \sqrt {b \,x^{2}+a}\, b^{3} x} \]
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 \[ \int {\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {7}{2}}\,{d x} \]
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 \[ \int {\left (c\,x\right )}^{7/2}\,{\left (b\,x^2+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 50.26, size = 46, normalized size = 0.22 \[ \frac {a^{\frac {3}{2}} c^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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