Optimal. Leaf size=198 \[ \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)} \]
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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} \[ \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)} \]
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 )^{3/2}}{(d+e x)^{3/2} (f+g x)^{11/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{9/2}}+\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 )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx}{9 (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 )^{5/2}}{9 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{9/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{7/2}}+\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 )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx}{63 (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 )^{5/2}}{9 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{9/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{7/2}}+\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 )^{5/2}}{315 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 105, normalized size = 0.53 \[ \frac {2 ((d+e x) (a e+c d x))^{5/2} \left (35 a^2 e^2 g^2-10 a c d e g (9 f+2 g x)+c^2 d^2 \left (63 f^2+36 f g x+8 g^2 x^2\right )\right )}{315 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.03, size = 918, normalized size = 4.64 \[ \frac {2 \, {\left (8 \, c^{4} d^{4} g^{2} x^{4} + 63 \, a^{2} c^{2} d^{2} e^{2} f^{2} - 90 \, a^{3} c d e^{3} f g + 35 \, a^{4} e^{4} g^{2} + 4 \, {\left (9 \, c^{4} d^{4} f g - a c^{3} d^{3} e g^{2}\right )} x^{3} + 3 \, {\left (21 \, c^{4} d^{4} f^{2} - 6 \, a c^{3} d^{3} e f g + a^{2} c^{2} d^{2} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (63 \, a c^{3} d^{3} e f^{2} - 72 \, a^{2} c^{2} d^{2} e^{2} f g + 25 \, a^{3} c d e^{3} 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}}{315 \, {\left (c^{3} d^{4} f^{8} - 3 \, a c^{2} d^{3} e f^{7} g + 3 \, a^{2} c d^{2} e^{2} f^{6} g^{2} - a^{3} d e^{3} f^{5} g^{3} + {\left (c^{3} d^{3} e f^{3} g^{5} - 3 \, a c^{2} d^{2} e^{2} f^{2} g^{6} + 3 \, a^{2} c d e^{3} f g^{7} - a^{3} e^{4} g^{8}\right )} x^{6} + {\left (5 \, c^{3} d^{3} e f^{4} g^{4} - a^{3} d e^{3} g^{8} + {\left (c^{3} d^{4} - 15 \, a c^{2} d^{2} e^{2}\right )} f^{3} g^{5} - 3 \, {\left (a c^{2} d^{3} e - 5 \, a^{2} c d e^{3}\right )} f^{2} g^{6} + {\left (3 \, a^{2} c d^{2} e^{2} - 5 \, a^{3} e^{4}\right )} f g^{7}\right )} x^{5} + 5 \, {\left (2 \, c^{3} d^{3} e f^{5} g^{3} - a^{3} d e^{3} f g^{7} + {\left (c^{3} d^{4} - 6 \, a c^{2} d^{2} e^{2}\right )} f^{4} g^{4} - 3 \, {\left (a c^{2} d^{3} e - 2 \, a^{2} c d e^{3}\right )} f^{3} g^{5} + {\left (3 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4}\right )} f^{2} g^{6}\right )} x^{4} + 10 \, {\left (c^{3} d^{3} e f^{6} g^{2} - a^{3} d e^{3} f^{2} g^{6} + {\left (c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} f^{5} g^{3} - 3 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{4} g^{4} + {\left (3 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{3} g^{5}\right )} x^{3} + 5 \, {\left (c^{3} d^{3} e f^{7} g - 2 \, a^{3} d e^{3} f^{3} g^{5} + {\left (2 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} f^{6} g^{2} - 3 \, {\left (2 \, a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{5} g^{3} + {\left (6 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{4} g^{4}\right )} x^{2} + {\left (c^{3} d^{3} e f^{8} - 5 \, a^{3} d e^{3} f^{4} g^{4} + {\left (5 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} f^{7} g - 3 \, {\left (5 \, a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{6} g^{2} + {\left (15 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{5} 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 \[ -\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}-20 a c d e \,g^{2} x +36 c^{2} d^{2} f g x +35 a^{2} e^{2} g^{2}-90 a c d e f g +63 f^{2} c^{2} d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{315 \left (g x +f \right )^{\frac {9}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.48, size = 377, normalized size = 1.90 \[ -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {70\,a^4\,e^4\,g^2-180\,a^3\,c\,d\,e^3\,f\,g+126\,a^2\,c^2\,d^2\,e^2\,f^2}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {x^2\,\left (6\,a^2\,c^2\,d^2\,e^2\,g^2-36\,a\,c^3\,d^3\,e\,f\,g+126\,c^4\,d^4\,f^2\right )}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c^4\,d^4\,x^4}{315\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^3}-\frac {8\,c^3\,d^3\,x^3\,\left (a\,e\,g-9\,c\,d\,f\right )}{315\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {4\,a\,c\,d\,e\,x\,\left (25\,a^2\,e^2\,g^2-72\,a\,c\,d\,e\,f\,g+63\,c^2\,d^2\,f^2\right )}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )}{x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}+\frac {4\,f\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {4\,f^3\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {6\,f^2\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}} \]
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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