Optimal. Leaf size=200 \[ -\frac {8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g) \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{693 c^3 d^3 e (d+e x)^{7/2}}+\frac {8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{99 c^2 d^2 e (d+e x)^{5/2}}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {870, 794, 648} \[ \frac {8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{99 c^2 d^2 e (d+e x)^{5/2}}-\frac {8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g) \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{693 c^3 d^3 e (d+e x)^{7/2}}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rule 870
Rubi steps
\begin {align*} \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=\frac {2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}}+\frac {(4 (c d f-a e g)) \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{11 c d}\\ &=\frac {8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 c^2 d^2 e (d+e x)^{5/2}}+\frac {2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}}+\frac {\left (4 (c d f-a e g) \left (9 f-\frac {7 d g}{e}-\frac {2 a e g}{c d}\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{99 c d}\\ &=\frac {8 (c d f-a e g) \left (9 f-\frac {7 d g}{e}-\frac {2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{693 c^2 d^2 (d+e x)^{7/2}}+\frac {8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 c^2 d^2 e (d+e x)^{5/2}}+\frac {2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 100, normalized size = 0.50 \[ \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (8 a^2 e^2 g^2-4 a c d e g (11 f+7 g x)+c^2 d^2 \left (99 f^2+154 f g x+63 g^2 x^2\right )\right )}{693 c^3 d^3 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 284, normalized size = 1.42 \[ \frac {2 \, {\left (63 \, c^{5} d^{5} g^{2} x^{5} + 99 \, a^{3} c^{2} d^{2} e^{3} f^{2} - 44 \, a^{4} c d e^{4} f g + 8 \, a^{5} e^{5} g^{2} + 7 \, {\left (22 \, c^{5} d^{5} f g + 23 \, a c^{4} d^{4} e g^{2}\right )} x^{4} + {\left (99 \, c^{5} d^{5} f^{2} + 418 \, a c^{4} d^{4} e f g + 113 \, a^{2} c^{3} d^{3} e^{2} g^{2}\right )} x^{3} + 3 \, {\left (99 \, a c^{4} d^{4} e f^{2} + 110 \, a^{2} c^{3} d^{3} e^{2} f g + a^{3} c^{2} d^{2} e^{3} g^{2}\right )} x^{2} + {\left (297 \, a^{2} c^{3} d^{3} e^{2} f^{2} + 22 \, a^{3} c^{2} d^{2} e^{3} f g - 4 \, a^{4} c d e^{4} g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{693 \, {\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 116, normalized size = 0.58 \[ \frac {2 \left (c d x +a e \right ) \left (63 g^{2} x^{2} c^{2} d^{2}-28 a c d e \,g^{2} x +154 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-44 a c d e f g +99 f^{2} c^{2} d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{693 \left (e x +d \right )^{\frac {5}{2}} c^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 243, normalized size = 1.22 \[ \frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{2}}{7 \, c d} + \frac {4 \, {\left (7 \, c^{4} d^{4} x^{4} + 19 \, a c^{3} d^{3} e x^{3} + 15 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{3} c d e^{3} x - 2 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f g}{63 \, c^{2} d^{2}} + \frac {2 \, {\left (63 \, c^{5} d^{5} x^{5} + 161 \, a c^{4} d^{4} e x^{4} + 113 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 3 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 4 \, a^{4} c d e^{4} x + 8 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} g^{2}}{693 \, c^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.56, size = 259, normalized size = 1.30 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {16\,a^5\,e^5\,g^2-88\,a^4\,c\,d\,e^4\,f\,g+198\,a^3\,c^2\,d^2\,e^3\,f^2}{693\,c^3\,d^3}+\frac {x^3\,\left (226\,a^2\,c^3\,d^3\,e^2\,g^2+836\,a\,c^4\,d^4\,e\,f\,g+198\,c^5\,d^5\,f^2\right )}{693\,c^3\,d^3}+\frac {2\,c^2\,d^2\,g^2\,x^5}{11}+\frac {2\,c\,d\,g\,x^4\,\left (23\,a\,e\,g+22\,c\,d\,f\right )}{99}+\frac {2\,a^2\,e^2\,x\,\left (-4\,a^2\,e^2\,g^2+22\,a\,c\,d\,e\,f\,g+297\,c^2\,d^2\,f^2\right )}{693\,c^2\,d^2}+\frac {2\,a\,e\,x^2\,\left (a^2\,e^2\,g^2+110\,a\,c\,d\,e\,f\,g+99\,c^2\,d^2\,f^2\right )}{231\,c\,d}\right )}{\sqrt {d+e\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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