Optimal. Leaf size=38 \[ \frac{2 \left (a+b \sqrt{x}\right )^7}{7 b^2}-\frac{a \left (a+b \sqrt{x}\right )^6}{3 b^2} \]
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Rubi [A] time = 0.0133622, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{2 \left (a+b \sqrt{x}\right )^7}{7 b^2}-\frac{a \left (a+b \sqrt{x}\right )^6}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt{x}\right )^5 \, dx &=2 \operatorname{Subst}\left (\int x (a+b x)^5 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^5}{b}+\frac{(a+b x)^6}{b}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a \left (a+b \sqrt{x}\right )^6}{3 b^2}+\frac{2 \left (a+b \sqrt{x}\right )^7}{7 b^2}\\ \end{align*}
Mathematica [A] time = 0.0197621, size = 28, normalized size = 0.74 \[ -\frac{\left (a-6 b \sqrt{x}\right ) \left (a+b \sqrt{x}\right )^6}{21 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 55, normalized size = 1.5 \begin{align*}{\frac{2\,{b}^{5}}{7}{x}^{{\frac{7}{2}}}}+{\frac{5\,a{b}^{4}{x}^{3}}{3}}+4\,{x}^{5/2}{a}^{2}{b}^{3}+5\,{a}^{3}{b}^{2}{x}^{2}+{\frac{10\,{a}^{4}b}{3}{x}^{{\frac{3}{2}}}}+x{a}^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94596, size = 73, normalized size = 1.92 \begin{align*} \frac{2}{7} \, b^{5} x^{\frac{7}{2}} + \frac{5}{3} \, a b^{4} x^{3} + 4 \, a^{2} b^{3} x^{\frac{5}{2}} + 5 \, a^{3} b^{2} x^{2} + \frac{10}{3} \, a^{4} b x^{\frac{3}{2}} + a^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44519, size = 130, normalized size = 3.42 \begin{align*} \frac{5}{3} \, a b^{4} x^{3} + 5 \, a^{3} b^{2} x^{2} + a^{5} x + \frac{2}{21} \,{\left (3 \, b^{5} x^{3} + 42 \, a^{2} b^{3} x^{2} + 35 \, a^{4} b x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.1036, size = 66, normalized size = 1.74 \begin{align*} a^{5} x + \frac{10 a^{4} b x^{\frac{3}{2}}}{3} + 5 a^{3} b^{2} x^{2} + 4 a^{2} b^{3} x^{\frac{5}{2}} + \frac{5 a b^{4} x^{3}}{3} + \frac{2 b^{5} x^{\frac{7}{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11843, size = 73, normalized size = 1.92 \begin{align*} \frac{2}{7} \, b^{5} x^{\frac{7}{2}} + \frac{5}{3} \, a b^{4} x^{3} + 4 \, a^{2} b^{3} x^{\frac{5}{2}} + 5 \, a^{3} b^{2} x^{2} + \frac{10}{3} \, a^{4} b x^{\frac{3}{2}} + a^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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