Optimal. Leaf size=198 \[ \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)} \]
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Rubi [A] time = 0.23, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {872, 860} \[ \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rule 872
Rubi steps
\begin {align*} \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} (f+g x)^{13/2}} \, dx &=\frac {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 f-a e g) (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {(4 c d) \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} (f+g x)^{11/2}} \, dx}{11 (c d f-a e g)}\\ &=\frac {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 f-a e g) (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {\left (8 c^2 d^2\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} (f+g x)^{9/2}} \, dx}{99 (c d f-a e g)^2}\\ &=\frac {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 f-a e g) (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{693 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 115, normalized size = 0.58 \[ \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (63 a^2 e^2 g^2-14 a c d e g (11 f+2 g x)+c^2 d^2 \left (99 f^2+44 f g x+8 g^2 x^2\right )\right )}{693 \sqrt {d+e x} (f+g x)^{11/2} (c d f-a e g)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 1101, normalized size = 5.56 \[ \frac {2 \, {\left (8 \, c^{5} d^{5} g^{2} x^{5} + 99 \, a^{3} c^{2} d^{2} e^{3} f^{2} - 154 \, a^{4} c d e^{4} f g + 63 \, a^{5} e^{5} g^{2} + 4 \, {\left (11 \, c^{5} d^{5} f g - a c^{4} d^{4} e g^{2}\right )} x^{4} + {\left (99 \, c^{5} d^{5} f^{2} - 22 \, a c^{4} d^{4} e f g + 3 \, a^{2} c^{3} d^{3} e^{2} g^{2}\right )} x^{3} + {\left (297 \, a c^{4} d^{4} e f^{2} - 330 \, a^{2} c^{3} d^{3} e^{2} f g + 113 \, 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} - 418 \, a^{3} c^{2} d^{2} e^{3} f g + 161 \, 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} \sqrt {g x + f}}{693 \, {\left (c^{3} d^{4} f^{9} - 3 \, a c^{2} d^{3} e f^{8} g + 3 \, a^{2} c d^{2} e^{2} f^{7} g^{2} - a^{3} d e^{3} f^{6} g^{3} + {\left (c^{3} d^{3} e f^{3} g^{6} - 3 \, a c^{2} d^{2} e^{2} f^{2} g^{7} + 3 \, a^{2} c d e^{3} f g^{8} - a^{3} e^{4} g^{9}\right )} x^{7} + {\left (6 \, c^{3} d^{3} e f^{4} g^{5} - a^{3} d e^{3} g^{9} + {\left (c^{3} d^{4} - 18 \, a c^{2} d^{2} e^{2}\right )} f^{3} g^{6} - 3 \, {\left (a c^{2} d^{3} e - 6 \, a^{2} c d e^{3}\right )} f^{2} g^{7} + 3 \, {\left (a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4}\right )} f g^{8}\right )} x^{6} + 3 \, {\left (5 \, c^{3} d^{3} e f^{5} g^{4} - 2 \, a^{3} d e^{3} f g^{8} + {\left (2 \, c^{3} d^{4} - 15 \, a c^{2} d^{2} e^{2}\right )} f^{4} g^{5} - 3 \, {\left (2 \, a c^{2} d^{3} e - 5 \, a^{2} c d e^{3}\right )} f^{3} g^{6} + {\left (6 \, a^{2} c d^{2} e^{2} - 5 \, a^{3} e^{4}\right )} f^{2} g^{7}\right )} x^{5} + 5 \, {\left (4 \, c^{3} d^{3} e f^{6} g^{3} - 3 \, a^{3} d e^{3} f^{2} g^{7} + 3 \, {\left (c^{3} d^{4} - 4 \, a c^{2} d^{2} e^{2}\right )} f^{5} g^{4} - 3 \, {\left (3 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} f^{4} g^{5} + {\left (9 \, a^{2} c d^{2} e^{2} - 4 \, a^{3} e^{4}\right )} f^{3} g^{6}\right )} x^{4} + 5 \, {\left (3 \, c^{3} d^{3} e f^{7} g^{2} - 4 \, a^{3} d e^{3} f^{3} g^{6} + {\left (4 \, c^{3} d^{4} - 9 \, a c^{2} d^{2} e^{2}\right )} f^{6} g^{3} - 3 \, {\left (4 \, a c^{2} d^{3} e - 3 \, a^{2} c d e^{3}\right )} f^{5} g^{4} + 3 \, {\left (4 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{4} g^{5}\right )} x^{3} + 3 \, {\left (2 \, c^{3} d^{3} e f^{8} g - 5 \, a^{3} d e^{3} f^{4} g^{5} + {\left (5 \, c^{3} d^{4} - 6 \, a c^{2} d^{2} e^{2}\right )} f^{7} g^{2} - 3 \, {\left (5 \, a c^{2} d^{3} e - 2 \, a^{2} c d e^{3}\right )} f^{6} g^{3} + {\left (15 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4}\right )} f^{5} g^{4}\right )} x^{2} + {\left (c^{3} d^{3} e f^{9} - 6 \, a^{3} d e^{3} f^{5} g^{4} + 3 \, {\left (2 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f^{8} g - 3 \, {\left (6 \, a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{7} g^{2} + {\left (18 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{6} g^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.01, size = 169, normalized size = 0.85 \[ -\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}-28 a c d e \,g^{2} x +44 c^{2} d^{2} f g x +63 a^{2} e^{2} g^{2}-154 a c d e f g +99 f^{2} c^{2} d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{693 \left (g x +f \right )^{\frac {11}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.82, size = 465, normalized size = 2.35 \[ -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {126\,a^5\,e^5\,g^2-308\,a^4\,c\,d\,e^4\,f\,g+198\,a^3\,c^2\,d^2\,e^3\,f^2}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {x^3\,\left (6\,a^2\,c^3\,d^3\,e^2\,g^2-44\,a\,c^4\,d^4\,e\,f\,g+198\,c^5\,d^5\,f^2\right )}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c^5\,d^5\,x^5}{693\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^3}-\frac {8\,c^4\,d^4\,x^4\,\left (a\,e\,g-11\,c\,d\,f\right )}{693\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {2\,a^2\,c\,d\,e^2\,x\,\left (161\,a^2\,e^2\,g^2-418\,a\,c\,d\,e\,f\,g+297\,c^2\,d^2\,f^2\right )}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {2\,a\,c^2\,d^2\,e\,x^2\,\left (113\,a^2\,e^2\,g^2-330\,a\,c\,d\,e\,f\,g+297\,c^2\,d^2\,f^2\right )}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )}{x^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^5}+\frac {5\,f\,x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {5\,f^4\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}+\frac {10\,f^2\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {10\,f^3\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}} \]
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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