Optimal. Leaf size=139 \[ \frac {5 a^7 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{1024 b^{9/2}}-\frac {5 a^5 (a+2 b x) \sqrt {a x+b x^2}}{1024 b^4}+\frac {5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac {a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac {\left (a x+b x^2\right )^{7/2}}{7 b} \]
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Rubi [A] time = 0.05, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {640, 612, 620, 206} \begin {gather*} -\frac {5 a^5 (a+2 b x) \sqrt {a x+b x^2}}{1024 b^4}+\frac {5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}+\frac {5 a^7 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{1024 b^{9/2}}-\frac {a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac {\left (a x+b x^2\right )^{7/2}}{7 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 640
Rubi steps
\begin {align*} \int x \left (a x+b x^2\right )^{5/2} \, dx &=\frac {\left (a x+b x^2\right )^{7/2}}{7 b}-\frac {a \int \left (a x+b x^2\right )^{5/2} \, dx}{2 b}\\ &=-\frac {a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac {\left (a x+b x^2\right )^{7/2}}{7 b}+\frac {\left (5 a^3\right ) \int \left (a x+b x^2\right )^{3/2} \, dx}{48 b^2}\\ &=\frac {5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac {a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac {\left (a x+b x^2\right )^{7/2}}{7 b}-\frac {\left (5 a^5\right ) \int \sqrt {a x+b x^2} \, dx}{256 b^3}\\ &=-\frac {5 a^5 (a+2 b x) \sqrt {a x+b x^2}}{1024 b^4}+\frac {5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac {a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac {\left (a x+b x^2\right )^{7/2}}{7 b}+\frac {\left (5 a^7\right ) \int \frac {1}{\sqrt {a x+b x^2}} \, dx}{2048 b^4}\\ &=-\frac {5 a^5 (a+2 b x) \sqrt {a x+b x^2}}{1024 b^4}+\frac {5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac {a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac {\left (a x+b x^2\right )^{7/2}}{7 b}+\frac {\left (5 a^7\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a x+b x^2}}\right )}{1024 b^4}\\ &=-\frac {5 a^5 (a+2 b x) \sqrt {a x+b x^2}}{1024 b^4}+\frac {5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac {a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac {\left (a x+b x^2\right )^{7/2}}{7 b}+\frac {5 a^7 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{1024 b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 131, normalized size = 0.94 \begin {gather*} \frac {\sqrt {x (a+b x)} \left (\frac {105 a^{13/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {x} \sqrt {\frac {b x}{a}+1}}+\sqrt {b} \left (-105 a^6+70 a^5 b x-56 a^4 b^2 x^2+48 a^3 b^3 x^3+4736 a^2 b^4 x^4+7424 a b^5 x^5+3072 b^6 x^6\right )\right )}{21504 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.43, size = 123, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a x+b x^2} \left (-105 a^6+70 a^5 b x-56 a^4 b^2 x^2+48 a^3 b^3 x^3+4736 a^2 b^4 x^4+7424 a b^5 x^5+3072 b^6 x^6\right )}{21504 b^4}-\frac {5 a^7 \log \left (-2 \sqrt {b} \sqrt {a x+b x^2}+a+2 b x\right )}{2048 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 236, normalized size = 1.70 \begin {gather*} \left [\frac {105 \, a^{7} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (3072 \, b^{7} x^{6} + 7424 \, a b^{6} x^{5} + 4736 \, a^{2} b^{5} x^{4} + 48 \, a^{3} b^{4} x^{3} - 56 \, a^{4} b^{3} x^{2} + 70 \, a^{5} b^{2} x - 105 \, a^{6} b\right )} \sqrt {b x^{2} + a x}}{43008 \, b^{5}}, -\frac {105 \, a^{7} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right ) - {\left (3072 \, b^{7} x^{6} + 7424 \, a b^{6} x^{5} + 4736 \, a^{2} b^{5} x^{4} + 48 \, a^{3} b^{4} x^{3} - 56 \, a^{4} b^{3} x^{2} + 70 \, a^{5} b^{2} x - 105 \, a^{6} b\right )} \sqrt {b x^{2} + a x}}{21504 \, b^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 120, normalized size = 0.86 \begin {gather*} -\frac {5 \, a^{7} \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right )}{2048 \, b^{\frac {9}{2}}} - \frac {1}{21504} \, \sqrt {b x^{2} + a x} {\left (\frac {105 \, a^{6}}{b^{4}} - 2 \, {\left (\frac {35 \, a^{5}}{b^{3}} - 4 \, {\left (\frac {7 \, a^{4}}{b^{2}} - 2 \, {\left (\frac {3 \, a^{3}}{b} + 8 \, {\left (37 \, a^{2} + 2 \, {\left (12 \, b^{2} x + 29 \, a b\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 165, normalized size = 1.19 \begin {gather*} \frac {5 a^{7} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2048 b^{\frac {9}{2}}}-\frac {5 \sqrt {b \,x^{2}+a x}\, a^{5} x}{512 b^{3}}-\frac {5 \sqrt {b \,x^{2}+a x}\, a^{6}}{1024 b^{4}}+\frac {5 \left (b \,x^{2}+a x \right )^{\frac {3}{2}} a^{3} x}{192 b^{2}}+\frac {5 \left (b \,x^{2}+a x \right )^{\frac {3}{2}} a^{4}}{384 b^{3}}-\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}} a x}{12 b}-\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}} a^{2}}{24 b^{2}}+\frac {\left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{7 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 163, normalized size = 1.17 \begin {gather*} -\frac {5 \, \sqrt {b x^{2} + a x} a^{5} x}{512 \, b^{3}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3} x}{192 \, b^{2}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} a x}{12 \, b} + \frac {5 \, a^{7} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2048 \, b^{\frac {9}{2}}} - \frac {5 \, \sqrt {b x^{2} + a x} a^{6}}{1024 \, b^{4}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4}}{384 \, b^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} a^{2}}{24 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {7}{2}}}{7 \, b} \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\,{\left (b\,x^2+a\,x\right )}^{5/2} \,d x \end {gather*}
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 \begin {gather*} \int x \left (x \left (a + b x\right )\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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