Optimal. Leaf size=223 \[ -\frac{1024 a^4 \sqrt{a x+b \sqrt{x}}}{143 b^6 x^{3/2}}+\frac{2560 a^3 \sqrt{a x+b \sqrt{x}}}{429 b^5 x^2}-\frac{2240 a^2 \sqrt{a x+b \sqrt{x}}}{429 b^4 x^{5/2}}-\frac{8192 a^6 \sqrt{a x+b \sqrt{x}}}{429 b^8 \sqrt{x}}+\frac{4096 a^5 \sqrt{a x+b \sqrt{x}}}{429 b^7 x}+\frac{672 a \sqrt{a x+b \sqrt{x}}}{143 b^3 x^3}-\frac{56 \sqrt{a x+b \sqrt{x}}}{13 b^2 x^{7/2}}+\frac{4}{b x^3 \sqrt{a x+b \sqrt{x}}} \]
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Rubi [A] time = 0.353146, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2015, 2016, 2014} \[ -\frac{1024 a^4 \sqrt{a x+b \sqrt{x}}}{143 b^6 x^{3/2}}+\frac{2560 a^3 \sqrt{a x+b \sqrt{x}}}{429 b^5 x^2}-\frac{2240 a^2 \sqrt{a x+b \sqrt{x}}}{429 b^4 x^{5/2}}-\frac{8192 a^6 \sqrt{a x+b \sqrt{x}}}{429 b^8 \sqrt{x}}+\frac{4096 a^5 \sqrt{a x+b \sqrt{x}}}{429 b^7 x}+\frac{672 a \sqrt{a x+b \sqrt{x}}}{143 b^3 x^3}-\frac{56 \sqrt{a x+b \sqrt{x}}}{13 b^2 x^{7/2}}+\frac{4}{b x^3 \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Rule 2015
Rule 2016
Rule 2014
Rubi steps
\begin{align*} \int \frac{1}{x^{7/2} \left (b \sqrt{x}+a x\right )^{3/2}} \, dx &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}+\frac{14 \int \frac{1}{x^4 \sqrt{b \sqrt{x}+a x}} \, dx}{b}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}-\frac{(168 a) \int \frac{1}{x^{7/2} \sqrt{b \sqrt{x}+a x}} \, dx}{13 b^2}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}+\frac{\left (1680 a^2\right ) \int \frac{1}{x^3 \sqrt{b \sqrt{x}+a x}} \, dx}{143 b^3}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}-\frac{2240 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^4 x^{5/2}}-\frac{\left (4480 a^3\right ) \int \frac{1}{x^{5/2} \sqrt{b \sqrt{x}+a x}} \, dx}{429 b^4}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}-\frac{2240 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^4 x^{5/2}}+\frac{2560 a^3 \sqrt{b \sqrt{x}+a x}}{429 b^5 x^2}+\frac{\left (1280 a^4\right ) \int \frac{1}{x^2 \sqrt{b \sqrt{x}+a x}} \, dx}{143 b^5}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}-\frac{2240 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^4 x^{5/2}}+\frac{2560 a^3 \sqrt{b \sqrt{x}+a x}}{429 b^5 x^2}-\frac{1024 a^4 \sqrt{b \sqrt{x}+a x}}{143 b^6 x^{3/2}}-\frac{\left (1024 a^5\right ) \int \frac{1}{x^{3/2} \sqrt{b \sqrt{x}+a x}} \, dx}{143 b^6}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}-\frac{2240 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^4 x^{5/2}}+\frac{2560 a^3 \sqrt{b \sqrt{x}+a x}}{429 b^5 x^2}-\frac{1024 a^4 \sqrt{b \sqrt{x}+a x}}{143 b^6 x^{3/2}}+\frac{4096 a^5 \sqrt{b \sqrt{x}+a x}}{429 b^7 x}+\frac{\left (2048 a^6\right ) \int \frac{1}{x \sqrt{b \sqrt{x}+a x}} \, dx}{429 b^7}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}-\frac{2240 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^4 x^{5/2}}+\frac{2560 a^3 \sqrt{b \sqrt{x}+a x}}{429 b^5 x^2}-\frac{1024 a^4 \sqrt{b \sqrt{x}+a x}}{143 b^6 x^{3/2}}+\frac{4096 a^5 \sqrt{b \sqrt{x}+a x}}{429 b^7 x}-\frac{8192 a^6 \sqrt{b \sqrt{x}+a x}}{429 b^8 \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0629794, size = 107, normalized size = 0.48 \[ -\frac{4 \left (-256 a^5 b^2 x^{5/2}+128 a^4 b^3 x^2-80 a^3 b^4 x^{3/2}+56 a^2 b^5 x+1024 a^6 b x^3+2048 a^7 x^{7/2}-42 a b^6 \sqrt{x}+33 b^7\right )}{429 b^8 x^3 \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 636, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} x^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.36309, size = 278, normalized size = 1.25 \begin{align*} \frac{4 \,{\left (1024 \, a^{7} b x^{4} - 384 \, a^{5} b^{3} x^{3} - 136 \, a^{3} b^{5} x^{2} - 75 \, a b^{7} x -{\left (2048 \, a^{8} x^{4} - 1280 \, a^{6} b^{2} x^{3} - 208 \, a^{4} b^{4} x^{2} - 98 \, a^{2} b^{6} x - 33 \, b^{8}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{429 \,{\left (a^{2} b^{8} x^{5} - b^{10} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} x^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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