Optimal. Leaf size=269 \[ -\frac {16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{3003 c^4 d^4 e (d+e x)^{7/2}}+\frac {16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2}{429 c^3 d^3 e (d+e x)^{5/2}}+\frac {12 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{143 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}} \]
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Rubi [A] time = 0.40, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {870, 794, 648} \[ \frac {12 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{143 c^2 d^2 (d+e x)^{7/2}}+\frac {16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2}{429 c^3 d^3 e (d+e x)^{5/2}}-\frac {16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{3003 c^4 d^4 e (d+e x)^{7/2}}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rule 870
Rubi steps
\begin {align*} \int \frac {(f+g x)^3 \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)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}}+\frac {(6 (c d f-a e g)) \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}{13 c d}\\ &=\frac {12 (c d f-a e g) (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}}{143 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}}+\frac {\left (24 (c d f-a e g)^2\right ) \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}{143 c^2 d^2}\\ &=\frac {16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 e (d+e x)^{5/2}}+\frac {12 (c d f-a e g) (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}}{143 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}}+\frac {\left (8 (c d f-a e g)^2 \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}{429 c^2 d^2}\\ &=\frac {16 (c d f-a e g)^2 \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}}{3003 c^3 d^3 (d+e x)^{7/2}}+\frac {16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 e (d+e x)^{5/2}}+\frac {12 (c d f-a e g) (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}}{143 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 147, normalized size = 0.55 \[ \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (-16 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (13 f+7 g x)-2 a c^2 d^2 e g \left (143 f^2+182 f g x+63 g^2 x^2\right )+c^3 d^3 \left (429 f^3+1001 f^2 g x+819 f g^2 x^2+231 g^3 x^3\right )\right )}{3003 c^4 d^4 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 416, normalized size = 1.55 \[ \frac {2 \, {\left (231 \, c^{6} d^{6} g^{3} x^{6} + 429 \, a^{3} c^{3} d^{3} e^{3} f^{3} - 286 \, a^{4} c^{2} d^{2} e^{4} f^{2} g + 104 \, a^{5} c d e^{5} f g^{2} - 16 \, a^{6} e^{6} g^{3} + 63 \, {\left (13 \, c^{6} d^{6} f g^{2} + 9 \, a c^{5} d^{5} e g^{3}\right )} x^{5} + 7 \, {\left (143 \, c^{6} d^{6} f^{2} g + 299 \, a c^{5} d^{5} e f g^{2} + 53 \, a^{2} c^{4} d^{4} e^{2} g^{3}\right )} x^{4} + {\left (429 \, c^{6} d^{6} f^{3} + 2717 \, a c^{5} d^{5} e f^{2} g + 1469 \, a^{2} c^{4} d^{4} e^{2} f g^{2} + 5 \, a^{3} c^{3} d^{3} e^{3} g^{3}\right )} x^{3} + 3 \, {\left (429 \, a c^{5} d^{5} e f^{3} + 715 \, a^{2} c^{4} d^{4} e^{2} f^{2} g + 13 \, a^{3} c^{3} d^{3} e^{3} f g^{2} - 2 \, a^{4} c^{2} d^{2} e^{4} g^{3}\right )} x^{2} + {\left (1287 \, a^{2} c^{4} d^{4} e^{2} f^{3} + 143 \, a^{3} c^{3} d^{3} e^{3} f^{2} g - 52 \, a^{4} c^{2} d^{2} e^{4} f g^{2} + 8 \, a^{5} c d e^{5} g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3003 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]
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 = 188, normalized size = 0.70 \[ -\frac {2 \left (c d x +a e \right ) \left (-231 g^{3} x^{3} c^{3} d^{3}+126 a \,c^{2} d^{2} e \,g^{3} x^{2}-819 c^{3} d^{3} f \,g^{2} x^{2}-56 a^{2} c d \,e^{2} g^{3} x +364 a \,c^{2} d^{2} e f \,g^{2} x -1001 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-104 a^{2} c d \,e^{2} f \,g^{2}+286 a \,c^{2} d^{2} e \,f^{2} g -429 f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 \left (e x +d \right )^{\frac {5}{2}} c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 362, normalized size = 1.35 \[ \frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{3}}{7 \, c d} + \frac {2 \, {\left (7 \, c^{4} d^{4} x^{4} + 19 \, a c^{3} d^{3} e x^{3} + 15 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{3} c d e^{3} x - 2 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f^{2} g}{21 \, c^{2} d^{2}} + \frac {2 \, {\left (63 \, c^{5} d^{5} x^{5} + 161 \, a c^{4} d^{4} e x^{4} + 113 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 3 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 4 \, a^{4} c d e^{4} x + 8 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} f g^{2}}{231 \, c^{3} d^{3}} + \frac {2 \, {\left (231 \, c^{6} d^{6} x^{6} + 567 \, a c^{5} d^{5} e x^{5} + 371 \, a^{2} c^{4} d^{4} e^{2} x^{4} + 5 \, a^{3} c^{3} d^{3} e^{3} x^{3} - 6 \, a^{4} c^{2} d^{2} e^{4} x^{2} + 8 \, a^{5} c d e^{5} x - 16 \, a^{6} e^{6}\right )} \sqrt {c d x + a e} g^{3}}{3003 \, c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.81, size = 379, normalized size = 1.41 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g\,x^4\,\left (53\,a^2\,e^2\,g^2+299\,a\,c\,d\,e\,f\,g+143\,c^2\,d^2\,f^2\right )}{429}-\frac {32\,a^6\,e^6\,g^3-208\,a^5\,c\,d\,e^5\,f\,g^2+572\,a^4\,c^2\,d^2\,e^4\,f^2\,g-858\,a^3\,c^3\,d^3\,e^3\,f^3}{3003\,c^4\,d^4}+\frac {x^3\,\left (10\,a^3\,c^3\,d^3\,e^3\,g^3+2938\,a^2\,c^4\,d^4\,e^2\,f\,g^2+5434\,a\,c^5\,d^5\,e\,f^2\,g+858\,c^6\,d^6\,f^3\right )}{3003\,c^4\,d^4}+\frac {2\,c^2\,d^2\,g^3\,x^6}{13}+\frac {6\,c\,d\,g^2\,x^5\,\left (9\,a\,e\,g+13\,c\,d\,f\right )}{143}+\frac {2\,a^2\,e^2\,x\,\left (8\,a^3\,e^3\,g^3-52\,a^2\,c\,d\,e^2\,f\,g^2+143\,a\,c^2\,d^2\,e\,f^2\,g+1287\,c^3\,d^3\,f^3\right )}{3003\,c^3\,d^3}+\frac {2\,a\,e\,x^2\,\left (-2\,a^3\,e^3\,g^3+13\,a^2\,c\,d\,e^2\,f\,g^2+715\,a\,c^2\,d^2\,e\,f^2\,g+429\,c^3\,d^3\,f^3\right )}{1001\,c^2\,d^2}\right )}{\sqrt {d+e\,x}} \]
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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