Optimal. Leaf size=116 \[ -\frac {5 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{3/2}}+\frac {5 a^3 \sqrt {x} \sqrt {a+b x}}{64 b}+\frac {5}{32} a^2 x^{3/2} \sqrt {a+b x}+\frac {5}{24} a x^{3/2} (a+b x)^{3/2}+\frac {1}{4} x^{3/2} (a+b x)^{5/2} \]
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Rubi [A] time = 0.04, antiderivative size = 116, 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 {5 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{3/2}}+\frac {5}{32} a^2 x^{3/2} \sqrt {a+b x}+\frac {5 a^3 \sqrt {x} \sqrt {a+b x}}{64 b}+\frac {5}{24} a x^{3/2} (a+b x)^{3/2}+\frac {1}{4} x^{3/2} (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 \sqrt {x} (a+b x)^{5/2} \, dx &=\frac {1}{4} x^{3/2} (a+b x)^{5/2}+\frac {1}{8} (5 a) \int \sqrt {x} (a+b x)^{3/2} \, dx\\ &=\frac {5}{24} a x^{3/2} (a+b x)^{3/2}+\frac {1}{4} x^{3/2} (a+b x)^{5/2}+\frac {1}{16} \left (5 a^2\right ) \int \sqrt {x} \sqrt {a+b x} \, dx\\ &=\frac {5}{32} a^2 x^{3/2} \sqrt {a+b x}+\frac {5}{24} a x^{3/2} (a+b x)^{3/2}+\frac {1}{4} x^{3/2} (a+b x)^{5/2}+\frac {1}{64} \left (5 a^3\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx\\ &=\frac {5 a^3 \sqrt {x} \sqrt {a+b x}}{64 b}+\frac {5}{32} a^2 x^{3/2} \sqrt {a+b x}+\frac {5}{24} a x^{3/2} (a+b x)^{3/2}+\frac {1}{4} x^{3/2} (a+b x)^{5/2}-\frac {\left (5 a^4\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{128 b}\\ &=\frac {5 a^3 \sqrt {x} \sqrt {a+b x}}{64 b}+\frac {5}{32} a^2 x^{3/2} \sqrt {a+b x}+\frac {5}{24} a x^{3/2} (a+b x)^{3/2}+\frac {1}{4} x^{3/2} (a+b x)^{5/2}-\frac {\left (5 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{64 b}\\ &=\frac {5 a^3 \sqrt {x} \sqrt {a+b x}}{64 b}+\frac {5}{32} a^2 x^{3/2} \sqrt {a+b x}+\frac {5}{24} a x^{3/2} (a+b x)^{3/2}+\frac {1}{4} x^{3/2} (a+b x)^{5/2}-\frac {\left (5 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}\\ &=\frac {5 a^3 \sqrt {x} \sqrt {a+b x}}{64 b}+\frac {5}{32} a^2 x^{3/2} \sqrt {a+b x}+\frac {5}{24} a x^{3/2} (a+b x)^{3/2}+\frac {1}{4} x^{3/2} (a+b x)^{5/2}-\frac {5 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 96, normalized size = 0.83 \begin {gather*} \frac {\sqrt {a+b x} \left (\sqrt {b} \sqrt {x} \left (15 a^3+118 a^2 b x+136 a b^2 x^2+48 b^3 x^3\right )-\frac {15 a^{7/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {\frac {b x}{a}+1}}\right )}{192 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 95, normalized size = 0.82 \begin {gather*} \frac {5 a^4 \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{64 b^{3/2}}+\frac {\sqrt {a+b x} \left (15 a^3 \sqrt {x}+118 a^2 b x^{3/2}+136 a b^2 x^{5/2}+48 b^3 x^{7/2}\right )}{192 b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.45, size = 162, normalized size = 1.40 \begin {gather*} \left [\frac {15 \, a^{4} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (48 \, b^{4} x^{3} + 136 \, a b^{3} x^{2} + 118 \, a^{2} b^{2} x + 15 \, a^{3} b\right )} \sqrt {b x + a} \sqrt {x}}{384 \, b^{2}}, \frac {15 \, a^{4} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (48 \, b^{4} x^{3} + 136 \, a b^{3} x^{2} + 118 \, a^{2} b^{2} x + 15 \, a^{3} b\right )} \sqrt {b x + a} \sqrt {x}}{192 \, b^{2}}\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 = 111, normalized size = 0.96 \begin {gather*} \frac {5 \sqrt {b x +a}\, a^{2} x^{\frac {3}{2}}}{32}-\frac {5 \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 {3}{2}} \sqrt {x}}+\frac {5 \sqrt {b x +a}\, a^{3} \sqrt {x}}{64 b}+\frac {5 \left (b x +a \right )^{\frac {3}{2}} a \,x^{\frac {3}{2}}}{24}+\frac {\left (b x +a \right )^{\frac {5}{2}} x^{\frac {3}{2}}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.98, size = 176, normalized size = 1.52 \begin {gather*} \frac {5 \, 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 {3}{2}}} + \frac {\frac {15 \, \sqrt {b x + a} a^{4} b^{3}}{\sqrt {x}} - \frac {55 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b^{2}}{x^{\frac {3}{2}}} + \frac {73 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} b}{x^{\frac {5}{2}}} + \frac {15 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4}}{x^{\frac {7}{2}}}}{192 \, {\left (b^{5} - \frac {4 \, {\left (b x + a\right )} b^{4}}{x} + \frac {6 \, {\left (b x + a\right )}^{2} b^{3}}{x^{2}} - \frac {4 \, {\left (b x + a\right )}^{3} b^{2}}{x^{3}} + \frac {{\left (b x + a\right )}^{4} b}{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 \sqrt {x}\,{\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 = 9.86, size = 155, normalized size = 1.34 \begin {gather*} \frac {5 a^{\frac {7}{2}} \sqrt {x}}{64 b \sqrt {1 + \frac {b x}{a}}} + \frac {133 a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 \sqrt {1 + \frac {b x}{a}}} + \frac {127 a^{\frac {3}{2}} b x^{\frac {5}{2}}}{96 \sqrt {1 + \frac {b x}{a}}} + \frac {23 \sqrt {a} b^{2} x^{\frac {7}{2}}}{24 \sqrt {1 + \frac {b x}{a}}} - \frac {5 a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {3}{2}}} + \frac {b^{3} 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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