Optimal. Leaf size=153 \[ -\frac {2 a^{7/4} c^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{21 b^{5/4} \sqrt {a+b x^2}}+\frac {2 (c x)^{5/2} \sqrt {a+b x^2}}{7 c}+\frac {4 a c \sqrt {c x} \sqrt {a+b x^2}}{21 b} \]
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Rubi [A] time = 0.09, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {279, 321, 329, 220} \[ -\frac {2 a^{7/4} c^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{21 b^{5/4} \sqrt {a+b x^2}}+\frac {2 (c x)^{5/2} \sqrt {a+b x^2}}{7 c}+\frac {4 a c \sqrt {c x} \sqrt {a+b x^2}}{21 b} \]
Antiderivative was successfully verified.
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Rule 220
Rule 279
Rule 321
Rule 329
Rubi steps
\begin {align*} \int (c x)^{3/2} \sqrt {a+b x^2} \, dx &=\frac {2 (c x)^{5/2} \sqrt {a+b x^2}}{7 c}+\frac {1}{7} (2 a) \int \frac {(c x)^{3/2}}{\sqrt {a+b x^2}} \, dx\\ &=\frac {4 a c \sqrt {c x} \sqrt {a+b x^2}}{21 b}+\frac {2 (c x)^{5/2} \sqrt {a+b x^2}}{7 c}-\frac {\left (2 a^2 c^2\right ) \int \frac {1}{\sqrt {c x} \sqrt {a+b x^2}} \, dx}{21 b}\\ &=\frac {4 a c \sqrt {c x} \sqrt {a+b x^2}}{21 b}+\frac {2 (c x)^{5/2} \sqrt {a+b x^2}}{7 c}-\frac {\left (4 a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{21 b}\\ &=\frac {4 a c \sqrt {c x} \sqrt {a+b x^2}}{21 b}+\frac {2 (c x)^{5/2} \sqrt {a+b x^2}}{7 c}-\frac {2 a^{7/4} c^{3/2} \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 )}{21 b^{5/4} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 85, normalized size = 0.56 \[ \frac {2 c \sqrt {c x} \sqrt {a+b x^2} \left (\left (a+b x^2\right ) \sqrt {\frac {b x^2}{a}+1}-a \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{7 b \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b x^{2} + a} \sqrt {c x} c x, x\right ) \]
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 \[ \int \sqrt {b x^{2} + a} \left (c x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 138, normalized size = 0.90 \[ -\frac {2 \sqrt {c x}\, \left (-3 b^{3} x^{5}-5 a \,b^{2} x^{3}-2 a^{2} b x +\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, a^{2} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )\right ) c}{21 \sqrt {b \,x^{2}+a}\, b^{2} x} \]
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 \[ \int \sqrt {b x^{2} + a} \left (c x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,x\right )}^{3/2}\,\sqrt {b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.76, size = 46, normalized size = 0.30 \[ \frac {\sqrt {a} c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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