Optimal. Leaf size=336 \[ -\frac {128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{15015 c^5 d^5 e (d+e x)^{5/2}}+\frac {128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^3}{3003 c^4 d^4 e (d+e x)^{3/2}}+\frac {32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{143 c^2 d^2 (d+e x)^{5/2}}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}} \]
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Rubi [A] time = 0.61, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {870, 794, 648} \[ \frac {16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{143 c^2 d^2 (d+e x)^{5/2}}+\frac {32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^3}{3003 c^4 d^4 e (d+e x)^{3/2}}-\frac {128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{15015 c^5 d^5 e (d+e x)^{5/2}}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rule 870
Rubi steps
\begin {align*} \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx &=\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}}+\frac {(8 (c d f-a e g)) \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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{13 c d}\\ &=\frac {16 (c d f-a e g) (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}}{143 c^2 d^2 (d+e x)^{5/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}}+\frac {\left (48 (c d f-a e g)^2\right ) \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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{143 c^2 d^2}\\ &=\frac {32 (c d f-a e g)^2 (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}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {16 (c d f-a e g) (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}}{143 c^2 d^2 (d+e x)^{5/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}}+\frac {\left (64 (c d f-a e g)^3\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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{429 c^3 d^3}\\ &=\frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3003 c^4 d^4 e (d+e x)^{3/2}}+\frac {32 (c d f-a e g)^2 (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}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {16 (c d f-a e g) (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}}{143 c^2 d^2 (d+e x)^{5/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}}+\frac {\left (64 (c d f-a e g)^3 \left (7 f-\frac {5 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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{3003 c^3 d^3}\\ &=\frac {128 (c d f-a e g)^3 \left (7 f-\frac {5 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 )^{5/2}}{15015 c^4 d^4 (d+e x)^{5/2}}+\frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3003 c^4 d^4 e (d+e x)^{3/2}}+\frac {32 (c d f-a e g)^2 (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}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {16 (c d f-a e g) (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}}{143 c^2 d^2 (d+e x)^{5/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 195, normalized size = 0.58 \[ \frac {2 ((d+e x) (a e+c d x))^{5/2} \left (128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (13 f+5 g x)+16 a^2 c^2 d^2 e^2 g^2 \left (143 f^2+130 f g x+35 g^2 x^2\right )-8 a c^3 d^3 e g \left (429 f^3+715 f^2 g x+455 f g^2 x^2+105 g^3 x^3\right )+c^4 d^4 \left (3003 f^4+8580 f^3 g x+10010 f^2 g^2 x^2+5460 f g^3 x^3+1155 g^4 x^4\right )\right )}{15015 c^5 d^5 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 472, normalized size = 1.40 \[ \frac {2 \, {\left (1155 \, c^{6} d^{6} g^{4} x^{6} + 3003 \, a^{2} c^{4} d^{4} e^{2} f^{4} - 3432 \, a^{3} c^{3} d^{3} e^{3} f^{3} g + 2288 \, a^{4} c^{2} d^{2} e^{4} f^{2} g^{2} - 832 \, a^{5} c d e^{5} f g^{3} + 128 \, a^{6} e^{6} g^{4} + 210 \, {\left (26 \, c^{6} d^{6} f g^{3} + 7 \, a c^{5} d^{5} e g^{4}\right )} x^{5} + 35 \, {\left (286 \, c^{6} d^{6} f^{2} g^{2} + 208 \, a c^{5} d^{5} e f g^{3} + a^{2} c^{4} d^{4} e^{2} g^{4}\right )} x^{4} + 20 \, {\left (429 \, c^{6} d^{6} f^{3} g + 715 \, a c^{5} d^{5} e f^{2} g^{2} + 13 \, a^{2} c^{4} d^{4} e^{2} f g^{3} - 2 \, a^{3} c^{3} d^{3} e^{3} g^{4}\right )} x^{3} + 3 \, {\left (1001 \, c^{6} d^{6} f^{4} + 4576 \, a c^{5} d^{5} e f^{3} g + 286 \, a^{2} c^{4} d^{4} e^{2} f^{2} g^{2} - 104 \, a^{3} c^{3} d^{3} e^{3} f g^{3} + 16 \, a^{4} c^{2} d^{2} e^{4} g^{4}\right )} x^{2} + 2 \, {\left (3003 \, a c^{5} d^{5} e f^{4} + 858 \, a^{2} c^{4} d^{4} e^{2} f^{3} g - 572 \, a^{3} c^{3} d^{3} e^{3} f^{2} g^{2} + 208 \, a^{4} c^{2} d^{2} e^{4} f g^{3} - 32 \, a^{5} c d e^{5} g^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{15015 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{4}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 283, normalized size = 0.84 \[ \frac {2 \left (c d x +a e \right ) \left (1155 g^{4} x^{4} c^{4} d^{4}-840 a \,c^{3} d^{3} e \,g^{4} x^{3}+5460 c^{4} d^{4} f \,g^{3} x^{3}+560 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-3640 a \,c^{3} d^{3} e f \,g^{3} x^{2}+10010 c^{4} d^{4} f^{2} g^{2} x^{2}-320 a^{3} c d \,e^{3} g^{4} x +2080 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -5720 a \,c^{3} d^{3} e \,f^{2} g^{2} x +8580 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-832 a^{3} c d \,e^{3} f \,g^{3}+2288 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-3432 a \,c^{3} d^{3} e \,f^{3} g +3003 f^{4} c^{4} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{15015 \left (e x +d \right )^{\frac {3}{2}} c^{5} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 413, normalized size = 1.23 \[ \frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt {c d x + a e} f^{4}}{5 \, c d} + \frac {8 \, {\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} e x^{2} + a^{2} c d e^{2} x - 2 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{3} g}{35 \, c^{2} d^{2}} + \frac {4 \, {\left (35 \, c^{4} d^{4} x^{4} + 50 \, a c^{3} d^{3} e x^{3} + 3 \, a^{2} c^{2} d^{2} e^{2} x^{2} - 4 \, a^{3} c d e^{3} x + 8 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f^{2} g^{2}}{105 \, c^{3} d^{3}} + \frac {8 \, {\left (105 \, c^{5} d^{5} x^{5} + 140 \, a c^{4} d^{4} e x^{4} + 5 \, a^{2} c^{3} d^{3} e^{2} x^{3} - 6 \, a^{3} c^{2} d^{2} e^{3} x^{2} + 8 \, a^{4} c d e^{4} x - 16 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} f g^{3}}{1155 \, c^{4} d^{4}} + \frac {2 \, {\left (1155 \, c^{6} d^{6} x^{6} + 1470 \, a c^{5} d^{5} e x^{5} + 35 \, a^{2} c^{4} d^{4} e^{2} x^{4} - 40 \, a^{3} c^{3} d^{3} e^{3} x^{3} + 48 \, a^{4} c^{2} d^{2} e^{4} x^{2} - 64 \, a^{5} c d e^{5} x + 128 \, a^{6} e^{6}\right )} \sqrt {c d x + a e} g^{4}}{15015 \, c^{5} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.80, size = 445, normalized size = 1.32 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,g^3\,x^5\,\left (7\,a\,e\,g+26\,c\,d\,f\right )}{143}+\frac {256\,a^6\,e^6\,g^4-1664\,a^5\,c\,d\,e^5\,f\,g^3+4576\,a^4\,c^2\,d^2\,e^4\,f^2\,g^2-6864\,a^3\,c^3\,d^3\,e^3\,f^3\,g+6006\,a^2\,c^4\,d^4\,e^2\,f^4}{15015\,c^5\,d^5}+\frac {x^2\,\left (96\,a^4\,c^2\,d^2\,e^4\,g^4-624\,a^3\,c^3\,d^3\,e^3\,f\,g^3+1716\,a^2\,c^4\,d^4\,e^2\,f^2\,g^2+27456\,a\,c^5\,d^5\,e\,f^3\,g+6006\,c^6\,d^6\,f^4\right )}{15015\,c^5\,d^5}+\frac {x\,\left (-128\,a^5\,c\,d\,e^5\,g^4+832\,a^4\,c^2\,d^2\,e^4\,f\,g^3-2288\,a^3\,c^3\,d^3\,e^3\,f^2\,g^2+3432\,a^2\,c^4\,d^4\,e^2\,f^3\,g+12012\,a\,c^5\,d^5\,e\,f^4\right )}{15015\,c^5\,d^5}+\frac {2\,c\,d\,g^4\,x^6}{13}+\frac {8\,g\,x^3\,\left (-2\,a^3\,e^3\,g^3+13\,a^2\,c\,d\,e^2\,f\,g^2+715\,a\,c^2\,d^2\,e\,f^2\,g+429\,c^3\,d^3\,f^3\right )}{3003\,c^2\,d^2}+\frac {2\,g^2\,x^4\,\left (a^2\,e^2\,g^2+208\,a\,c\,d\,e\,f\,g+286\,c^2\,d^2\,f^2\right )}{429\,c\,d}\right )}{\sqrt {d+e\,x}} \]
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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