Optimal. Leaf size=118 \[ -\frac {5 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{512 b^{7/2}}+\frac {5 a^4 (a+2 b x) \sqrt {a x+b x^2}}{512 b^3}-\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b} \]
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Rubi [A] time = 0.04, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {612, 620, 206} \[ \frac {5 a^4 (a+2 b x) \sqrt {a x+b x^2}}{512 b^3}-\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}-\frac {5 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{512 b^{7/2}}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rubi steps
\begin {align*} \int \left (a x+b x^2\right )^{5/2} \, dx &=\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {\left (5 a^2\right ) \int \left (a x+b x^2\right )^{3/2} \, dx}{24 b}\\ &=-\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}+\frac {\left (5 a^4\right ) \int \sqrt {a x+b x^2} \, dx}{128 b^2}\\ &=\frac {5 a^4 (a+2 b x) \sqrt {a x+b x^2}}{512 b^3}-\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {\left (5 a^6\right ) \int \frac {1}{\sqrt {a x+b x^2}} \, dx}{1024 b^3}\\ &=\frac {5 a^4 (a+2 b x) \sqrt {a x+b x^2}}{512 b^3}-\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {\left (5 a^6\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a x+b x^2}}\right )}{512 b^3}\\ &=\frac {5 a^4 (a+2 b x) \sqrt {a x+b x^2}}{512 b^3}-\frac {5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {5 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{512 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 120, normalized size = 1.02 \[ \frac {\sqrt {x (a+b x)} \left (\sqrt {b} \left (15 a^5-10 a^4 b x+8 a^3 b^2 x^2+432 a^2 b^3 x^3+640 a b^4 x^4+256 b^5 x^5\right )-\frac {15 a^{11/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {x} \sqrt {\frac {b x}{a}+1}}\right )}{1536 b^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 213, normalized size = 1.81 \[ \left [\frac {15 \, a^{6} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt {b x^{2} + a x}}{3072 \, b^{4}}, \frac {15 \, a^{6} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right ) + {\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt {b x^{2} + a x}}{1536 \, b^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 107, normalized size = 0.91 \[ \frac {5 \, a^{6} \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right )}{1024 \, b^{\frac {7}{2}}} + \frac {1}{1536} \, \sqrt {b x^{2} + a x} {\left (\frac {15 \, a^{5}}{b^{3}} - 2 \, {\left (\frac {5 \, a^{4}}{b^{2}} - 4 \, {\left (\frac {a^{3}}{b} + 2 \, {\left (27 \, a^{2} + 8 \, {\left (2 \, b^{2} x + 5 \, a b\right )} x\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 134, normalized size = 1.14 \[ -\frac {5 a^{6} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{1024 b^{\frac {7}{2}}}+\frac {5 \sqrt {b \,x^{2}+a x}\, a^{4} x}{256 b^{2}}+\frac {5 \sqrt {b \,x^{2}+a x}\, a^{5}}{512 b^{3}}-\frac {5 \left (b \,x^{2}+a x \right )^{\frac {3}{2}} a^{2} x}{96 b}-\frac {5 \left (b \,x^{2}+a x \right )^{\frac {3}{2}} a^{3}}{192 b^{2}}+\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{12 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 141, normalized size = 1.19 \[ \frac {1}{6} \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} x + \frac {5 \, \sqrt {b x^{2} + a x} a^{4} x}{256 \, b^{2}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} x}{96 \, b} - \frac {5 \, a^{6} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{1024 \, b^{\frac {7}{2}}} + \frac {5 \, \sqrt {b x^{2} + a x} a^{5}}{512 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}}{192 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} a}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.56, size = 119, normalized size = 1.01 \[ \frac {{\left (b\,x^2+a\,x\right )}^{5/2}\,\left (\frac {a}{2}+b\,x\right )}{6\,b}-\frac {5\,a^2\,\left (\frac {{\left (b\,x^2+a\,x\right )}^{3/2}\,\left (\frac {a}{2}+b\,x\right )}{4\,b}-\frac {3\,a^2\,\left (\sqrt {b\,x^2+a\,x}\,\left (\frac {x}{2}+\frac {a}{4\,b}\right )-\frac {a^2\,\ln \left (\frac {\frac {a}{2}+b\,x}{\sqrt {b}}+\sqrt {b\,x^2+a\,x}\right )}{8\,b^{3/2}}\right )}{16\,b}\right )}{24\,b} \]
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 \[ \int \left (a x + b x^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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