Optimal. Leaf size=321 \[ -\frac{2 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g+c d (e f-7 d g)\right )}{35 c^2 d^2 g \sqrt{d+e x}}-\frac{8 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right )}{105 c^3 d^3 e}+\frac{8 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{105 c^4 d^4 e g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d g \sqrt{d+e x}} \]
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Rubi [A] time = 0.419845, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {880, 870, 794, 648} \[ -\frac{2 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g+c d (e f-7 d g)\right )}{35 c^2 d^2 g \sqrt{d+e x}}-\frac{8 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right )}{105 c^3 d^3 e}+\frac{8 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{105 c^4 d^4 e g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d g \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 880
Rule 870
Rule 794
Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2} (f+g x)^2}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 e (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt{d+e x}}-\frac{1}{7} \left (-7 d+\frac{6 a e^2}{c d}+\frac{e f}{g}\right ) \int \frac{\sqrt{d+e x} (f+g x)^2}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac{2 \left (6 a e^2 g+c d (e f-7 d g)\right ) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt{d+e x}}-\frac{\left (4 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right )\right ) \int \frac{\sqrt{d+e x} (f+g x)}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 c^2 d^2 g}\\ &=-\frac{8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 e}-\frac{2 \left (6 a e^2 g+c d (e f-7 d g)\right ) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt{d+e x}}+\frac{\left (4 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{105 c^3 d^3 e g}\\ &=\frac{8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^4 d^4 e g \sqrt{d+e x}}-\frac{8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 e}-\frac{2 \left (6 a e^2 g+c d (e f-7 d g)\right ) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.18984, size = 169, normalized size = 0.53 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (8 a^2 c d e^2 g (7 d g+14 e f+3 e g x)-48 a^3 e^4 g^2-2 a c^2 d^2 e \left (14 d g (5 f+g x)+e \left (35 f^2+28 f g x+9 g^2 x^2\right )\right )+c^3 d^3 \left (7 d \left (15 f^2+10 f g x+3 g^2 x^2\right )+e x \left (35 f^2+42 f g x+15 g^2 x^2\right )\right )\right )}{105 c^4 d^4 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 255, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -15\,e{g}^{2}{x}^{3}{c}^{3}{d}^{3}+18\,a{c}^{2}{d}^{2}{e}^{2}{g}^{2}{x}^{2}-21\,{c}^{3}{d}^{4}{g}^{2}{x}^{2}-42\,{c}^{3}{d}^{3}efg{x}^{2}-24\,{a}^{2}cd{e}^{3}{g}^{2}x+28\,a{c}^{2}{d}^{3}e{g}^{2}x+56\,a{c}^{2}{d}^{2}{e}^{2}fgx-70\,{c}^{3}{d}^{4}fgx-35\,{c}^{3}{d}^{3}e{f}^{2}x+48\,{a}^{3}{e}^{4}{g}^{2}-56\,{a}^{2}c{d}^{2}{e}^{2}{g}^{2}-112\,{a}^{2}cd{e}^{3}fg+140\,a{c}^{2}{d}^{3}efg+70\,a{c}^{2}{d}^{2}{e}^{2}{f}^{2}-105\,{d}^{4}{f}^{2}{c}^{3} \right ) }{105\,{c}^{4}{d}^{4}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45183, size = 417, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{2}}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} + \frac{4 \,{\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} +{\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} -{\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f g}{15 \, \sqrt{c d x + a e} c^{3} d^{3}} + \frac{2 \,{\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \,{\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} -{\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} g^{2}}{105 \, \sqrt{c d x + a e} c^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65022, size = 533, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (15 \, c^{3} d^{3} e g^{2} x^{3} + 35 \,{\left (3 \, c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2}\right )} f^{2} - 28 \,{\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} f g + 8 \,{\left (7 \, a^{2} c d^{2} e^{2} - 6 \, a^{3} e^{4}\right )} g^{2} + 3 \,{\left (14 \, c^{3} d^{3} e f g +{\left (7 \, c^{3} d^{4} - 6 \, a c^{2} d^{2} e^{2}\right )} g^{2}\right )} x^{2} +{\left (35 \, c^{3} d^{3} e f^{2} + 14 \,{\left (5 \, c^{3} d^{4} - 4 \, a c^{2} d^{2} e^{2}\right )} f g - 4 \,{\left (7 \, a c^{2} d^{3} e - 6 \, a^{2} c d e^{3}\right )} g^{2}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{105 \,{\left (c^{4} d^{4} e x + c^{4} d^{5}\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{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{2}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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