Optimal. Leaf size=140 \[ \frac {3 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{5/2}}-\frac {3 a^4 \sqrt {x} \sqrt {a+b x}}{128 b^2}+\frac {a^3 x^{3/2} \sqrt {a+b x}}{64 b}+\frac {1}{16} a^2 x^{5/2} \sqrt {a+b x}+\frac {1}{8} a x^{5/2} (a+b x)^{3/2}+\frac {1}{5} x^{5/2} (a+b x)^{5/2} \]
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Rubi [A] time = 0.05, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 8, 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^4 \sqrt {x} \sqrt {a+b x}}{128 b^2}+\frac {3 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{5/2}}+\frac {a^3 x^{3/2} \sqrt {a+b x}}{64 b}+\frac {1}{16} a^2 x^{5/2} \sqrt {a+b x}+\frac {1}{8} a x^{5/2} (a+b x)^{3/2}+\frac {1}{5} x^{5/2} (a+b x)^{5/2} \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)^{5/2} \, dx &=\frac {1}{5} x^{5/2} (a+b x)^{5/2}+\frac {1}{2} a \int x^{3/2} (a+b x)^{3/2} \, dx\\ &=\frac {1}{8} a x^{5/2} (a+b x)^{3/2}+\frac {1}{5} x^{5/2} (a+b x)^{5/2}+\frac {1}{16} \left (3 a^2\right ) \int x^{3/2} \sqrt {a+b x} \, dx\\ &=\frac {1}{16} a^2 x^{5/2} \sqrt {a+b x}+\frac {1}{8} a x^{5/2} (a+b x)^{3/2}+\frac {1}{5} x^{5/2} (a+b x)^{5/2}+\frac {1}{32} a^3 \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx\\ &=\frac {a^3 x^{3/2} \sqrt {a+b x}}{64 b}+\frac {1}{16} a^2 x^{5/2} \sqrt {a+b x}+\frac {1}{8} a x^{5/2} (a+b x)^{3/2}+\frac {1}{5} x^{5/2} (a+b x)^{5/2}-\frac {\left (3 a^4\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{128 b}\\ &=-\frac {3 a^4 \sqrt {x} \sqrt {a+b x}}{128 b^2}+\frac {a^3 x^{3/2} \sqrt {a+b x}}{64 b}+\frac {1}{16} a^2 x^{5/2} \sqrt {a+b x}+\frac {1}{8} a x^{5/2} (a+b x)^{3/2}+\frac {1}{5} x^{5/2} (a+b x)^{5/2}+\frac {\left (3 a^5\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{256 b^2}\\ &=-\frac {3 a^4 \sqrt {x} \sqrt {a+b x}}{128 b^2}+\frac {a^3 x^{3/2} \sqrt {a+b x}}{64 b}+\frac {1}{16} a^2 x^{5/2} \sqrt {a+b x}+\frac {1}{8} a x^{5/2} (a+b x)^{3/2}+\frac {1}{5} x^{5/2} (a+b x)^{5/2}+\frac {\left (3 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{128 b^2}\\ &=-\frac {3 a^4 \sqrt {x} \sqrt {a+b x}}{128 b^2}+\frac {a^3 x^{3/2} \sqrt {a+b x}}{64 b}+\frac {1}{16} a^2 x^{5/2} \sqrt {a+b x}+\frac {1}{8} a x^{5/2} (a+b x)^{3/2}+\frac {1}{5} x^{5/2} (a+b x)^{5/2}+\frac {\left (3 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^2}\\ &=-\frac {3 a^4 \sqrt {x} \sqrt {a+b x}}{128 b^2}+\frac {a^3 x^{3/2} \sqrt {a+b x}}{64 b}+\frac {1}{16} a^2 x^{5/2} \sqrt {a+b x}+\frac {1}{8} a x^{5/2} (a+b x)^{3/2}+\frac {1}{5} x^{5/2} (a+b x)^{5/2}+\frac {3 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 107, normalized size = 0.76 \begin {gather*} \frac {\sqrt {a+b x} \left (\frac {15 a^{9/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {\frac {b x}{a}+1}}+\sqrt {b} \sqrt {x} \left (-15 a^4+10 a^3 b x+248 a^2 b^2 x^2+336 a b^3 x^3+128 b^4 x^4\right )\right )}{640 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 108, normalized size = 0.77 \begin {gather*} \frac {\sqrt {a+b x} \left (-15 a^4 \sqrt {x}+10 a^3 b x^{3/2}+248 a^2 b^2 x^{5/2}+336 a b^3 x^{7/2}+128 b^4 x^{9/2}\right )}{640 b^2}-\frac {3 a^5 \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{128 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 185, normalized size = 1.32 \begin {gather*} \left [\frac {15 \, a^{5} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (128 \, b^{5} x^{4} + 336 \, a b^{4} x^{3} + 248 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} \sqrt {b x + a} \sqrt {x}}{1280 \, b^{3}}, -\frac {15 \, a^{5} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (128 \, b^{5} x^{4} + 336 \, a b^{4} x^{3} + 248 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} \sqrt {b x + a} \sqrt {x}}{640 \, 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 = 138, normalized size = 0.99 \begin {gather*} \frac {3 \sqrt {\left (b x +a \right ) x}\, a^{5} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{256 \sqrt {b x +a}\, b^{\frac {5}{2}} \sqrt {x}}+\frac {3 \sqrt {b x +a}\, a^{4} \sqrt {x}}{128 b^{2}}+\frac {\left (b x +a \right )^{\frac {3}{2}} a^{3} \sqrt {x}}{64 b^{2}}+\frac {\left (b x +a \right )^{\frac {5}{2}} a^{2} \sqrt {x}}{80 b^{2}}+\frac {\left (b x +a \right )^{\frac {7}{2}} x^{\frac {3}{2}}}{5 b}-\frac {3 \left (b x +a \right )^{\frac {7}{2}} a \sqrt {x}}{40 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.98, size = 212, normalized size = 1.51 \begin {gather*} -\frac {3 \, a^{5} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{256 \, b^{\frac {5}{2}}} - \frac {\frac {15 \, \sqrt {b x + a} a^{5} b^{4}}{\sqrt {x}} - \frac {70 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} b^{3}}{x^{\frac {3}{2}}} + \frac {128 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} b^{2}}{x^{\frac {5}{2}}} + \frac {70 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{5} b}{x^{\frac {7}{2}}} - \frac {15 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{5}}{x^{\frac {9}{2}}}}{640 \, {\left (b^{7} - \frac {5 \, {\left (b x + a\right )} b^{6}}{x} + \frac {10 \, {\left (b x + a\right )}^{2} b^{5}}{x^{2}} - \frac {10 \, {\left (b x + a\right )}^{3} b^{4}}{x^{3}} + \frac {5 \, {\left (b x + a\right )}^{4} b^{3}}{x^{4}} - \frac {{\left (b x + a\right )}^{5} b^{2}}{x^{5}}\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 )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.41, size = 180, normalized size = 1.29 \begin {gather*} - \frac {3 a^{\frac {9}{2}} \sqrt {x}}{128 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {a^{\frac {7}{2}} x^{\frac {3}{2}}}{128 b \sqrt {1 + \frac {b x}{a}}} + \frac {129 a^{\frac {5}{2}} x^{\frac {5}{2}}}{320 \sqrt {1 + \frac {b x}{a}}} + \frac {73 a^{\frac {3}{2}} b x^{\frac {7}{2}}}{80 \sqrt {1 + \frac {b x}{a}}} + \frac {29 \sqrt {a} b^{2} x^{\frac {9}{2}}}{40 \sqrt {1 + \frac {b x}{a}}} + \frac {3 a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{128 b^{\frac {5}{2}}} + \frac {b^{3} x^{\frac {11}{2}}}{5 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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