Optimal. Leaf size=267 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)} \]
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Rubi [A] time = 0.308033, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {872, 860} \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 872
Rule 860
Rubi steps
\begin{align*} \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{11/2}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\frac{(2 c d) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{9/2}} \, dx}{3 (c d f-a e g)}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\frac{4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{7/2}}+\frac{\left (8 c^2 d^2\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{7/2}} \, dx}{21 (c d f-a e g)^2}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\frac{4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{7/2}}+\frac{16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac{\left (16 c^3 d^3\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{5/2}} \, dx}{105 (c d f-a e g)^3}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\frac{4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{7/2}}+\frac{16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac{32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 (c d f-a e g)^4 (d+e x)^{3/2} (f+g x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.154941, size = 152, normalized size = 0.57 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (15 a^2 c d e^2 g^2 (9 f+2 g x)-35 a^3 e^3 g^3-3 a c^2 d^2 e g \left (63 f^2+36 f g x+8 g^2 x^2\right )+c^3 d^3 \left (126 f^2 g x+105 f^3+72 f g^2 x^2+16 g^3 x^3\right )\right )}{315 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 260, normalized size = 1. \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{c}^{3}{d}^{3}{g}^{3}{x}^{3}+24\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-72\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-30\,{a}^{2}cd{e}^{2}{g}^{3}x+108\,a{c}^{2}{d}^{2}ef{g}^{2}x-126\,{c}^{3}{d}^{3}{f}^{2}gx+35\,{a}^{3}{e}^{3}{g}^{3}-135\,{a}^{2}cd{e}^{2}f{g}^{2}+189\,a{c}^{2}{d}^{2}e{f}^{2}g-105\,{c}^{3}{d}^{3}{f}^{3} \right ) }{315\,{g}^{4}{e}^{4}{a}^{4}-1260\,cd{g}^{3}f{e}^{3}{a}^{3}+1890\,{c}^{2}{d}^{2}{g}^{2}{f}^{2}{e}^{2}{a}^{2}-1260\,{c}^{3}{d}^{3}g{f}^{3}ea+315\,{c}^{4}{d}^{4}{f}^{4}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( gx+f \right ) ^{-{\frac{9}{2}}}{\frac{1}{\sqrt{ex+d}}}} \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{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{\sqrt{e x + d}{\left (g x + f\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99459, size = 2323, normalized size = 8.7 \begin{align*} \text{result too large to display} \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{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{\sqrt{e x + d}{\left (g x + f\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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