Optimal. Leaf size=269 \[ -\frac {16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{1155 c^4 d^4 e (d+e x)^{5/2}}+\frac {16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2}{231 c^3 d^3 e (d+e x)^{3/2}}+\frac {4 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{33 c^2 d^2 (d+e x)^{5/2}}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}} \]
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Rubi [A] time = 0.41, 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} \begin {gather*} \frac {4 (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)}{33 c^2 d^2 (d+e x)^{5/2}}+\frac {16 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)^2}{231 c^3 d^3 e (d+e x)^{3/2}}-\frac {16 \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 \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{1155 c^4 d^4 e (d+e x)^{5/2}}+\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 )^{5/2}}{11 c d (d+e x)^{5/2}} \end {gather*}
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 )^{3/2}}{(d+e x)^{3/2}} \, 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 )^{5/2}}{11 c d (d+e x)^{5/2}}+\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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{11 c d}\\ &=\frac {4 (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 )^{5/2}}{33 c^2 d^2 (d+e x)^{5/2}}+\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 )^{5/2}}{11 c d (d+e x)^{5/2}}+\frac {\left (8 (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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{33 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 )^{5/2}}{231 c^3 d^3 e (d+e x)^{3/2}}+\frac {4 (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 )^{5/2}}{33 c^2 d^2 (d+e x)^{5/2}}+\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 )^{5/2}}{11 c d (d+e x)^{5/2}}+\frac {\left (8 (c d f-a e g)^2 \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}{231 c^2 d^2}\\ &=\frac {16 (c d f-a e g)^2 \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}}{1155 c^3 d^3 (d+e x)^{5/2}}+\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 )^{5/2}}{231 c^3 d^3 e (d+e x)^{3/2}}+\frac {4 (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 )^{5/2}}{33 c^2 d^2 (d+e x)^{5/2}}+\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 )^{5/2}}{11 c d (d+e x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 137, normalized size = 0.51 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{5/2} \left (-16 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (11 f+5 g x)-2 a c^2 d^2 e g \left (99 f^2+110 f g x+35 g^2 x^2\right )+c^3 d^3 \left (231 f^3+495 f^2 g x+385 f g^2 x^2+105 g^3 x^3\right )\right )}{1155 c^4 d^4 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 340, normalized size = 1.26 \begin {gather*} \frac {2 \, {\left (105 \, c^{5} d^{5} g^{3} x^{5} + 231 \, a^{2} c^{3} d^{3} e^{2} f^{3} - 198 \, a^{3} c^{2} d^{2} e^{3} f^{2} g + 88 \, a^{4} c d e^{4} f g^{2} - 16 \, a^{5} e^{5} g^{3} + 35 \, {\left (11 \, c^{5} d^{5} f g^{2} + 4 \, a c^{4} d^{4} e g^{3}\right )} x^{4} + 5 \, {\left (99 \, c^{5} d^{5} f^{2} g + 110 \, a c^{4} d^{4} e f g^{2} + a^{2} c^{3} d^{3} e^{2} g^{3}\right )} x^{3} + 3 \, {\left (77 \, c^{5} d^{5} f^{3} + 264 \, a c^{4} d^{4} e f^{2} g + 11 \, a^{2} c^{3} d^{3} e^{2} f g^{2} - 2 \, a^{3} c^{2} d^{2} e^{3} g^{3}\right )} x^{2} + {\left (462 \, a c^{4} d^{4} e f^{3} + 99 \, a^{2} c^{3} d^{3} e^{2} f^{2} g - 44 \, a^{3} c^{2} d^{2} e^{3} f g^{2} + 8 \, a^{4} c d e^{4} 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}}{1155 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \end {gather*}
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 \begin {gather*} \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 )}^{3}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
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 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-105 g^{3} x^{3} c^{3} d^{3}+70 a \,c^{2} d^{2} e \,g^{3} x^{2}-385 c^{3} d^{3} f \,g^{2} x^{2}-40 a^{2} c d \,e^{2} g^{3} x +220 a \,c^{2} d^{2} e f \,g^{2} x -495 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-88 a^{2} c d \,e^{2} f \,g^{2}+198 a \,c^{2} d^{2} e \,f^{2} g -231 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 {3}{2}}}{1155 \left (e x +d \right )^{\frac {3}{2}} c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 294, normalized size = 1.09 \begin {gather*} \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^{3}}{5 \, c d} + \frac {6 \, {\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^{2} g}{35 \, c^{2} d^{2}} + \frac {2 \, {\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 g^{2}}{105 \, c^{3} d^{3}} + \frac {2 \, {\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} g^{3}}{1155 \, c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.65, size = 310, normalized size = 1.15 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^2\,x^4\,\left (4\,a\,e\,g+11\,c\,d\,f\right )}{33}-\frac {32\,a^5\,e^5\,g^3-176\,a^4\,c\,d\,e^4\,f\,g^2+396\,a^3\,c^2\,d^2\,e^3\,f^2\,g-462\,a^2\,c^3\,d^3\,e^2\,f^3}{1155\,c^4\,d^4}+\frac {x^2\,\left (-12\,a^3\,c^2\,d^2\,e^3\,g^3+66\,a^2\,c^3\,d^3\,e^2\,f\,g^2+1584\,a\,c^4\,d^4\,e\,f^2\,g+462\,c^5\,d^5\,f^3\right )}{1155\,c^4\,d^4}+\frac {2\,c\,d\,g^3\,x^5}{11}+\frac {2\,g\,x^3\,\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}+\frac {2\,a\,e\,x\,\left (8\,a^3\,e^3\,g^3-44\,a^2\,c\,d\,e^2\,f\,g^2+99\,a\,c^2\,d^2\,e\,f^2\,g+462\,c^3\,d^3\,f^3\right )}{1155\,c^3\,d^3}\right )}{\sqrt {d+e\,x}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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