Optimal. Leaf size=119 \[ \frac {3 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{5/2}}-\frac {3 a^3 \sqrt {x} \sqrt {a+b x}}{64 b^2}+\frac {a^2 x^{3/2} \sqrt {a+b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a+b x}+\frac {1}{4} x^{5/2} (a+b x)^{3/2} \]
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Rubi [A] time = 0.04, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {50, 63, 217, 206} \begin {gather*} -\frac {3 a^3 \sqrt {x} \sqrt {a+b x}}{64 b^2}+\frac {3 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{5/2}}+\frac {a^2 x^{3/2} \sqrt {a+b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a+b x}+\frac {1}{4} x^{5/2} (a+b x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int x^{3/2} (a+b x)^{3/2} \, dx &=\frac {1}{4} x^{5/2} (a+b x)^{3/2}+\frac {1}{8} (3 a) \int x^{3/2} \sqrt {a+b x} \, dx\\ &=\frac {1}{8} a x^{5/2} \sqrt {a+b x}+\frac {1}{4} x^{5/2} (a+b x)^{3/2}+\frac {1}{16} a^2 \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx\\ &=\frac {a^2 x^{3/2} \sqrt {a+b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a+b x}+\frac {1}{4} x^{5/2} (a+b x)^{3/2}-\frac {\left (3 a^3\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{64 b}\\ &=-\frac {3 a^3 \sqrt {x} \sqrt {a+b x}}{64 b^2}+\frac {a^2 x^{3/2} \sqrt {a+b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a+b x}+\frac {1}{4} x^{5/2} (a+b x)^{3/2}+\frac {\left (3 a^4\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{128 b^2}\\ &=-\frac {3 a^3 \sqrt {x} \sqrt {a+b x}}{64 b^2}+\frac {a^2 x^{3/2} \sqrt {a+b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a+b x}+\frac {1}{4} x^{5/2} (a+b x)^{3/2}+\frac {\left (3 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{64 b^2}\\ &=-\frac {3 a^3 \sqrt {x} \sqrt {a+b x}}{64 b^2}+\frac {a^2 x^{3/2} \sqrt {a+b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a+b x}+\frac {1}{4} x^{5/2} (a+b x)^{3/2}+\frac {\left (3 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^2}\\ &=-\frac {3 a^3 \sqrt {x} \sqrt {a+b x}}{64 b^2}+\frac {a^2 x^{3/2} \sqrt {a+b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a+b x}+\frac {1}{4} x^{5/2} (a+b x)^{3/2}+\frac {3 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 96, normalized size = 0.81 \begin {gather*} \frac {\sqrt {a+b x} \left (\frac {3 a^{7/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {\frac {b x}{a}+1}}+\sqrt {b} \sqrt {x} \left (-3 a^3+2 a^2 b x+24 a b^2 x^2+16 b^3 x^3\right )\right )}{64 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 95, normalized size = 0.80 \begin {gather*} \frac {\sqrt {a+b x} \left (-3 a^3 \sqrt {x}+2 a^2 b x^{3/2}+24 a b^2 x^{5/2}+16 b^3 x^{7/2}\right )}{64 b^2}-\frac {3 a^4 \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{64 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 163, normalized size = 1.37 \begin {gather*} \left [\frac {3 \, a^{4} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (16 \, b^{4} x^{3} + 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x - 3 \, a^{3} b\right )} \sqrt {b x + a} \sqrt {x}}{128 \, b^{3}}, -\frac {3 \, a^{4} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (16 \, b^{4} x^{3} + 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x - 3 \, a^{3} b\right )} \sqrt {b x + a} \sqrt {x}}{64 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 120, normalized size = 1.01 \begin {gather*} \frac {3 \sqrt {\left (b x +a \right ) x}\, a^{4} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{128 \sqrt {b x +a}\, b^{\frac {5}{2}} \sqrt {x}}+\frac {3 \sqrt {b x +a}\, a^{3} \sqrt {x}}{64 b^{2}}+\frac {\left (b x +a \right )^{\frac {3}{2}} a^{2} \sqrt {x}}{32 b^{2}}+\frac {\left (b x +a \right )^{\frac {5}{2}} x^{\frac {3}{2}}}{4 b}-\frac {\left (b x +a \right )^{\frac {5}{2}} a \sqrt {x}}{8 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.96, size = 178, normalized size = 1.50 \begin {gather*} -\frac {3 \, a^{4} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{128 \, b^{\frac {5}{2}}} - \frac {\frac {3 \, \sqrt {b x + a} a^{4} b^{3}}{\sqrt {x}} - \frac {11 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b^{2}}{x^{\frac {3}{2}}} - \frac {11 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} b}{x^{\frac {5}{2}}} + \frac {3 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4}}{x^{\frac {7}{2}}}}{64 \, {\left (b^{6} - \frac {4 \, {\left (b x + a\right )} b^{5}}{x} + \frac {6 \, {\left (b x + a\right )}^{2} b^{4}}{x^{2}} - \frac {4 \, {\left (b x + a\right )}^{3} b^{3}}{x^{3}} + \frac {{\left (b x + a\right )}^{4} b^{2}}{x^{4}}\right )}} \end {gather*}
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 \begin {gather*} \int x^{3/2}\,{\left (a+b\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.28, size = 153, normalized size = 1.29 \begin {gather*} - \frac {3 a^{\frac {7}{2}} \sqrt {x}}{64 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {a^{\frac {5}{2}} x^{\frac {3}{2}}}{64 b \sqrt {1 + \frac {b x}{a}}} + \frac {13 a^{\frac {3}{2}} x^{\frac {5}{2}}}{32 \sqrt {1 + \frac {b x}{a}}} + \frac {5 \sqrt {a} b x^{\frac {7}{2}}}{8 \sqrt {1 + \frac {b x}{a}}} + \frac {3 a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {5}{2}}} + \frac {b^{2} x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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