Optimal. Leaf size=122 \[ \frac{35 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{4 b^{9/2}}-\frac{14 x^3}{3 b^2 \sqrt{a x+b x^2}}+\frac{35 x \sqrt{a x+b x^2}}{6 b^3}-\frac{35 a \sqrt{a x+b x^2}}{4 b^4}-\frac{2 x^5}{3 b \left (a x+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0582204, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {668, 670, 640, 620, 206} \[ \frac{35 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{4 b^{9/2}}-\frac{14 x^3}{3 b^2 \sqrt{a x+b x^2}}+\frac{35 x \sqrt{a x+b x^2}}{6 b^3}-\frac{35 a \sqrt{a x+b x^2}}{4 b^4}-\frac{2 x^5}{3 b \left (a x+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 668
Rule 670
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^6}{\left (a x+b x^2\right )^{5/2}} \, dx &=-\frac{2 x^5}{3 b \left (a x+b x^2\right )^{3/2}}+\frac{7 \int \frac{x^4}{\left (a x+b x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac{2 x^5}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{14 x^3}{3 b^2 \sqrt{a x+b x^2}}+\frac{35 \int \frac{x^2}{\sqrt{a x+b x^2}} \, dx}{3 b^2}\\ &=-\frac{2 x^5}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{14 x^3}{3 b^2 \sqrt{a x+b x^2}}+\frac{35 x \sqrt{a x+b x^2}}{6 b^3}-\frac{(35 a) \int \frac{x}{\sqrt{a x+b x^2}} \, dx}{4 b^3}\\ &=-\frac{2 x^5}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{14 x^3}{3 b^2 \sqrt{a x+b x^2}}-\frac{35 a \sqrt{a x+b x^2}}{4 b^4}+\frac{35 x \sqrt{a x+b x^2}}{6 b^3}+\frac{\left (35 a^2\right ) \int \frac{1}{\sqrt{a x+b x^2}} \, dx}{8 b^4}\\ &=-\frac{2 x^5}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{14 x^3}{3 b^2 \sqrt{a x+b x^2}}-\frac{35 a \sqrt{a x+b x^2}}{4 b^4}+\frac{35 x \sqrt{a x+b x^2}}{6 b^3}+\frac{\left (35 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )}{4 b^4}\\ &=-\frac{2 x^5}{3 b \left (a x+b x^2\right )^{3/2}}-\frac{14 x^3}{3 b^2 \sqrt{a x+b x^2}}-\frac{35 a \sqrt{a x+b x^2}}{4 b^4}+\frac{35 x \sqrt{a x+b x^2}}{6 b^3}+\frac{35 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{4 b^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0145022, size = 50, normalized size = 0.41 \[ \frac{2 x^5 \sqrt{\frac{b x}{a}+1} \, _2F_1\left (\frac{5}{2},\frac{9}{2};\frac{11}{2};-\frac{b x}{a}\right )}{9 a^2 \sqrt{x (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 176, normalized size = 1.4 \begin{align*}{\frac{{x}^{5}}{2\,b} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,a{x}^{4}}{4\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{a}^{2}{x}^{3}}{24\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{x}^{2}{a}^{3}}{16\,{b}^{4}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{a}^{4}x}{48\,{b}^{5}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{245\,{a}^{2}x}{24\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}-{\frac{35\,{a}^{3}}{48\,{b}^{5}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}+{\frac{35\,{a}^{2}}{8}\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98898, size = 556, normalized size = 4.56 \begin{align*} \left [\frac{105 \,{\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )} \sqrt{b} \log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (6 \, b^{4} x^{3} - 21 \, a b^{3} x^{2} - 140 \, a^{2} b^{2} x - 105 \, a^{3} b\right )} \sqrt{b x^{2} + a x}}{24 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac{105 \,{\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (6 \, b^{4} x^{3} - 21 \, a b^{3} x^{2} - 140 \, a^{2} b^{2} x - 105 \, a^{3} b\right )} \sqrt{b x^{2} + a x}}{12 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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