Optimal. Leaf size=107 \[ \frac {3 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{128 b^{5/2}}-\frac {3 a^3 (a+2 b x) \sqrt {a x+b x^2}}{128 b^2}+\frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2} \]
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Rubi [A] time = 0.04, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {664, 612, 620, 206} \[ -\frac {3 a^3 (a+2 b x) \sqrt {a x+b x^2}}{128 b^2}+\frac {3 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{128 b^{5/2}}+\frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 664
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^2\right )^{5/2}}{x} \, dx &=\frac {1}{5} \left (a x+b x^2\right )^{5/2}+\frac {1}{2} a \int \left (a x+b x^2\right )^{3/2} \, dx\\ &=\frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2}-\frac {\left (3 a^3\right ) \int \sqrt {a x+b x^2} \, dx}{32 b}\\ &=-\frac {3 a^3 (a+2 b x) \sqrt {a x+b x^2}}{128 b^2}+\frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2}+\frac {\left (3 a^5\right ) \int \frac {1}{\sqrt {a x+b x^2}} \, dx}{256 b^2}\\ &=-\frac {3 a^3 (a+2 b x) \sqrt {a x+b x^2}}{128 b^2}+\frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2}+\frac {\left (3 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a x+b x^2}}\right )}{128 b^2}\\ &=-\frac {3 a^3 (a+2 b x) \sqrt {a x+b x^2}}{128 b^2}+\frac {a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac {1}{5} \left (a x+b x^2\right )^{5/2}+\frac {3 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{128 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 109, normalized size = 1.02 \[ \frac {\sqrt {x (a+b x)} \left (\frac {15 a^{9/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {x} \sqrt {\frac {b x}{a}+1}}+\sqrt {b} \left (-15 a^4+10 a^3 b x+248 a^2 b^2 x^2+336 a b^3 x^3+128 b^4 x^4\right )\right )}{640 b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 192, normalized size = 1.79 \[ \left [\frac {15 \, a^{5} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (128 \, b^{5} x^{4} + 336 \, a b^{4} x^{3} + 248 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} \sqrt {b x^{2} + a x}}{1280 \, b^{3}}, -\frac {15 \, a^{5} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right ) - {\left (128 \, b^{5} x^{4} + 336 \, a b^{4} x^{3} + 248 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} \sqrt {b x^{2} + a x}}{640 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 96, normalized size = 0.90 \[ -\frac {3 \, a^{5} \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right )}{256 \, b^{\frac {5}{2}}} - \frac {1}{640} \, \sqrt {b x^{2} + a x} {\left (\frac {15 \, a^{4}}{b^{2}} - 2 \, {\left (\frac {5 \, a^{3}}{b} + 4 \, {\left (31 \, a^{2} + 2 \, {\left (8 \, b^{2} x + 21 \, a b\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 120, normalized size = 1.12 \[ \frac {3 a^{5} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{256 b^{\frac {5}{2}}}-\frac {3 \sqrt {b \,x^{2}+a x}\, a^{3} x}{64 b}-\frac {3 \sqrt {b \,x^{2}+a x}\, a^{4}}{128 b^{2}}+\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}} a x}{8}+\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}} a^{2}}{16 b}+\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 118, normalized size = 1.10 \[ \frac {1}{8} \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x - \frac {3 \, \sqrt {b x^{2} + a x} a^{3} x}{64 \, b} + \frac {3 \, a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {5}{2}}} + \frac {1}{5} \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} - \frac {3 \, \sqrt {b x^{2} + a x} a^{4}}{128 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}}{16 \, b} \]
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 \[ \int \frac {{\left (b\,x^2+a\,x\right )}^{5/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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