Optimal. Leaf size=192 \[ 2002 a^{10} b^5 x^{3/2}+\frac{5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac{3003}{4} a^5 b^{10} x^4+\frac{910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac{210}{11} a^2 b^{13} x^{11/2}+910 a^{12} b^3 \sqrt{x}+1365 a^{11} b^4 x+105 a^{13} b^2 \log (x)-\frac{30 a^{14} b}{\sqrt{x}}-\frac{a^{15}}{x}+\frac{5}{2} a b^{14} x^6+\frac{2}{13} b^{15} x^{13/2} \]
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Rubi [A] time = 0.117044, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ 2002 a^{10} b^5 x^{3/2}+\frac{5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac{3003}{4} a^5 b^{10} x^4+\frac{910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac{210}{11} a^2 b^{13} x^{11/2}+910 a^{12} b^3 \sqrt{x}+1365 a^{11} b^4 x+105 a^{13} b^2 \log (x)-\frac{30 a^{14} b}{\sqrt{x}}-\frac{a^{15}}{x}+\frac{5}{2} a b^{14} x^6+\frac{2}{13} b^{15} x^{13/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (455 a^{12} b^3+\frac{a^{15}}{x^3}+\frac{15 a^{14} b}{x^2}+\frac{105 a^{13} b^2}{x}+1365 a^{11} b^4 x+3003 a^{10} b^5 x^2+5005 a^9 b^6 x^3+6435 a^8 b^7 x^4+6435 a^7 b^8 x^5+5005 a^6 b^9 x^6+3003 a^5 b^{10} x^7+1365 a^4 b^{11} x^8+455 a^3 b^{12} x^9+105 a^2 b^{13} x^{10}+15 a b^{14} x^{11}+b^{15} x^{12}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{15}}{x}-\frac{30 a^{14} b}{\sqrt{x}}+910 a^{12} b^3 \sqrt{x}+1365 a^{11} b^4 x+2002 a^{10} b^5 x^{3/2}+\frac{5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac{3003}{4} a^5 b^{10} x^4+\frac{910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac{210}{11} a^2 b^{13} x^{11/2}+\frac{5}{2} a b^{14} x^6+\frac{2}{13} b^{15} x^{13/2}+105 a^{13} b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0945768, size = 192, normalized size = 1. \[ 2002 a^{10} b^5 x^{3/2}+\frac{5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac{3003}{4} a^5 b^{10} x^4+\frac{910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac{210}{11} a^2 b^{13} x^{11/2}+910 a^{12} b^3 \sqrt{x}+1365 a^{11} b^4 x+105 a^{13} b^2 \log (x)-\frac{30 a^{14} b}{\sqrt{x}}-\frac{a^{15}}{x}+\frac{5}{2} a b^{14} x^6+\frac{2}{13} b^{15} x^{13/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 165, normalized size = 0.9 \begin{align*} -{\frac{{a}^{15}}{x}}+1365\,{a}^{11}{b}^{4}x+2002\,{a}^{10}{b}^{5}{x}^{3/2}+{\frac{5005\,{a}^{9}{b}^{6}{x}^{2}}{2}}+2574\,{a}^{8}{b}^{7}{x}^{5/2}+2145\,{a}^{7}{b}^{8}{x}^{3}+1430\,{a}^{6}{b}^{9}{x}^{7/2}+{\frac{3003\,{a}^{5}{b}^{10}{x}^{4}}{4}}+{\frac{910\,{a}^{4}{b}^{11}}{3}{x}^{{\frac{9}{2}}}}+91\,{a}^{3}{b}^{12}{x}^{5}+{\frac{210\,{a}^{2}{b}^{13}}{11}{x}^{{\frac{11}{2}}}}+{\frac{5\,a{b}^{14}{x}^{6}}{2}}+{\frac{2\,{b}^{15}}{13}{x}^{{\frac{13}{2}}}}+105\,{a}^{13}{b}^{2}\ln \left ( x \right ) -30\,{\frac{{a}^{14}b}{\sqrt{x}}}+910\,{a}^{12}{b}^{3}\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957969, size = 223, normalized size = 1.16 \begin{align*} \frac{2}{13} \, b^{15} x^{\frac{13}{2}} + \frac{5}{2} \, a b^{14} x^{6} + \frac{210}{11} \, a^{2} b^{13} x^{\frac{11}{2}} + 91 \, a^{3} b^{12} x^{5} + \frac{910}{3} \, a^{4} b^{11} x^{\frac{9}{2}} + \frac{3003}{4} \, a^{5} b^{10} x^{4} + 1430 \, a^{6} b^{9} x^{\frac{7}{2}} + 2145 \, a^{7} b^{8} x^{3} + 2574 \, a^{8} b^{7} x^{\frac{5}{2}} + \frac{5005}{2} \, a^{9} b^{6} x^{2} + 2002 \, a^{10} b^{5} x^{\frac{3}{2}} + 1365 \, a^{11} b^{4} x + 105 \, a^{13} b^{2} \log \left (x\right ) + 910 \, a^{12} b^{3} \sqrt{x} - \frac{30 \, a^{14} b \sqrt{x} + a^{15}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31613, size = 473, normalized size = 2.46 \begin{align*} \frac{4290 \, a b^{14} x^{7} + 156156 \, a^{3} b^{12} x^{6} + 1288287 \, a^{5} b^{10} x^{5} + 3680820 \, a^{7} b^{8} x^{4} + 4294290 \, a^{9} b^{6} x^{3} + 2342340 \, a^{11} b^{4} x^{2} + 360360 \, a^{13} b^{2} x \log \left (\sqrt{x}\right ) - 1716 \, a^{15} + 8 \,{\left (33 \, b^{15} x^{7} + 4095 \, a^{2} b^{13} x^{6} + 65065 \, a^{4} b^{11} x^{5} + 306735 \, a^{6} b^{9} x^{4} + 552123 \, a^{8} b^{7} x^{3} + 429429 \, a^{10} b^{5} x^{2} + 195195 \, a^{12} b^{3} x - 6435 \, a^{14} b\right )} \sqrt{x}}{1716 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.16904, size = 197, normalized size = 1.03 \begin{align*} - \frac{a^{15}}{x} - \frac{30 a^{14} b}{\sqrt{x}} + 105 a^{13} b^{2} \log{\left (x \right )} + 910 a^{12} b^{3} \sqrt{x} + 1365 a^{11} b^{4} x + 2002 a^{10} b^{5} x^{\frac{3}{2}} + \frac{5005 a^{9} b^{6} x^{2}}{2} + 2574 a^{8} b^{7} x^{\frac{5}{2}} + 2145 a^{7} b^{8} x^{3} + 1430 a^{6} b^{9} x^{\frac{7}{2}} + \frac{3003 a^{5} b^{10} x^{4}}{4} + \frac{910 a^{4} b^{11} x^{\frac{9}{2}}}{3} + 91 a^{3} b^{12} x^{5} + \frac{210 a^{2} b^{13} x^{\frac{11}{2}}}{11} + \frac{5 a b^{14} x^{6}}{2} + \frac{2 b^{15} x^{\frac{13}{2}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10184, size = 224, normalized size = 1.17 \begin{align*} \frac{2}{13} \, b^{15} x^{\frac{13}{2}} + \frac{5}{2} \, a b^{14} x^{6} + \frac{210}{11} \, a^{2} b^{13} x^{\frac{11}{2}} + 91 \, a^{3} b^{12} x^{5} + \frac{910}{3} \, a^{4} b^{11} x^{\frac{9}{2}} + \frac{3003}{4} \, a^{5} b^{10} x^{4} + 1430 \, a^{6} b^{9} x^{\frac{7}{2}} + 2145 \, a^{7} b^{8} x^{3} + 2574 \, a^{8} b^{7} x^{\frac{5}{2}} + \frac{5005}{2} \, a^{9} b^{6} x^{2} + 2002 \, a^{10} b^{5} x^{\frac{3}{2}} + 1365 \, a^{11} b^{4} x + 105 \, a^{13} b^{2} \log \left ({\left | x \right |}\right ) + 910 \, a^{12} b^{3} \sqrt{x} - \frac{30 \, a^{14} b \sqrt{x} + a^{15}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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