Optimal. Leaf size=267 \[ \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)} \]
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Rubi [A] time = 0.31, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {872, 860} \begin {gather*} \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^4}+\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 )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 (d+e x)^{3/2} (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 )^{3/2}}{9 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 860
Rule 872
Rubi steps
\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (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 )^{3/2}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\frac {(2 c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx}{3 (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 )^{3/2}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {\left (8 c^2 d^2\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx}{21 (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 )^{3/2}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 (c d f-a e g)^2 (d+e x)^{3/2} (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 )^{3/2}}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {\left (16 c^3 d^3\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx}{105 (c d f-a e g)^3}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{9/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 (c d f-a e g)^2 (d+e x)^{3/2} (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 )^{3/2}}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 (c d f-a e g)^4 (d+e x)^{3/2} (f+g x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 152, normalized size = 0.57 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{3/2} \left (-35 a^3 e^3 g^3+15 a^2 c d e^2 g^2 (9 f+2 g x)-3 a c^2 d^2 e g \left (63 f^2+36 f g x+8 g^2 x^2\right )+c^3 d^3 \left (105 f^3+126 f^2 g x+72 f g^2 x^2+16 g^3 x^3\right )\right )}{315 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 8.88, size = 169, normalized size = 0.63 \begin {gather*} \frac {2 \sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {105 c^3 d^3 (a e+c d x)^{3/2}}{(f+g x)^{3/2}}-\frac {189 c^2 d^2 g (a e+c d x)^{5/2}}{(f+g x)^{5/2}}-\frac {35 g^3 (a e+c d x)^{9/2}}{(f+g x)^{9/2}}+\frac {135 c d g^2 (a e+c d x)^{7/2}}{(f+g x)^{7/2}}\right )}{315 \sqrt {(d+e x) (a e+c d x)} (c d f-a e g)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 1179, normalized size = 4.42 \begin {gather*} \frac {2 \, {\left (16 \, c^{4} d^{4} g^{3} x^{4} + 105 \, a c^{3} d^{3} e f^{3} - 189 \, a^{2} c^{2} d^{2} e^{2} f^{2} g + 135 \, a^{3} c d e^{3} f g^{2} - 35 \, a^{4} e^{4} g^{3} + 8 \, {\left (9 \, c^{4} d^{4} f g^{2} - a c^{3} d^{3} e g^{3}\right )} x^{3} + 6 \, {\left (21 \, c^{4} d^{4} f^{2} g - 6 \, a c^{3} d^{3} e f g^{2} + a^{2} c^{2} d^{2} e^{2} g^{3}\right )} x^{2} + {\left (105 \, c^{4} d^{4} f^{3} - 63 \, a c^{3} d^{3} e f^{2} g + 27 \, a^{2} c^{2} d^{2} e^{2} f g^{2} - 5 \, a^{3} c d e^{3} 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} \sqrt {g x + f}}{315 \, {\left (c^{4} d^{5} f^{9} - 4 \, a c^{3} d^{4} e f^{8} g + 6 \, a^{2} c^{2} d^{3} e^{2} f^{7} g^{2} - 4 \, a^{3} c d^{2} e^{3} f^{6} g^{3} + a^{4} d e^{4} f^{5} g^{4} + {\left (c^{4} d^{4} e f^{4} g^{5} - 4 \, a c^{3} d^{3} e^{2} f^{3} g^{6} + 6 \, a^{2} c^{2} d^{2} e^{3} f^{2} g^{7} - 4 \, a^{3} c d e^{4} f g^{8} + a^{4} e^{5} g^{9}\right )} x^{6} + {\left (5 \, c^{4} d^{4} e f^{5} g^{4} + a^{4} d e^{4} g^{9} + {\left (c^{4} d^{5} - 20 \, a c^{3} d^{3} e^{2}\right )} f^{4} g^{5} - 2 \, {\left (2 \, a c^{3} d^{4} e - 15 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{3} g^{6} + 2 \, {\left (3 \, a^{2} c^{2} d^{3} e^{2} - 10 \, a^{3} c d e^{4}\right )} f^{2} g^{7} - {\left (4 \, a^{3} c d^{2} e^{3} - 5 \, a^{4} e^{5}\right )} f g^{8}\right )} x^{5} + 5 \, {\left (2 \, c^{4} d^{4} e f^{6} g^{3} + a^{4} d e^{4} f g^{8} + {\left (c^{4} d^{5} - 8 \, a c^{3} d^{3} e^{2}\right )} f^{5} g^{4} - 4 \, {\left (a c^{3} d^{4} e - 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{4} g^{5} + 2 \, {\left (3 \, a^{2} c^{2} d^{3} e^{2} - 4 \, a^{3} c d e^{4}\right )} f^{3} g^{6} - 2 \, {\left (2 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{2} g^{7}\right )} x^{4} + 10 \, {\left (c^{4} d^{4} e f^{7} g^{2} + a^{4} d e^{4} f^{2} g^{7} + {\left (c^{4} d^{5} - 4 \, a c^{3} d^{3} e^{2}\right )} f^{6} g^{3} - 2 \, {\left (2 \, a c^{3} d^{4} e - 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{5} g^{4} + 2 \, {\left (3 \, a^{2} c^{2} d^{3} e^{2} - 2 \, a^{3} c d e^{4}\right )} f^{4} g^{5} - {\left (4 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{3} g^{6}\right )} x^{3} + 5 \, {\left (c^{4} d^{4} e f^{8} g + 2 \, a^{4} d e^{4} f^{3} g^{6} + 2 \, {\left (c^{4} d^{5} - 2 \, a c^{3} d^{3} e^{2}\right )} f^{7} g^{2} - 2 \, {\left (4 \, a c^{3} d^{4} e - 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{6} g^{3} + 4 \, {\left (3 \, a^{2} c^{2} d^{3} e^{2} - a^{3} c d e^{4}\right )} f^{5} g^{4} - {\left (8 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{4} g^{5}\right )} x^{2} + {\left (c^{4} d^{4} e f^{9} + 5 \, a^{4} d e^{4} f^{4} g^{5} + {\left (5 \, c^{4} d^{5} - 4 \, a c^{3} d^{3} e^{2}\right )} f^{8} g - 2 \, {\left (10 \, a c^{3} d^{4} e - 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{7} g^{2} + 2 \, {\left (15 \, a^{2} c^{2} d^{3} e^{2} - 2 \, a^{3} c d e^{4}\right )} f^{6} g^{3} - {\left (20 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{5} g^{4}\right )} x\right )}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 260, normalized size = 0.97 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+24 a \,c^{2} d^{2} e \,g^{3} x^{2}-72 c^{3} d^{3} f \,g^{2} x^{2}-30 a^{2} c d \,e^{2} g^{3} x +108 a \,c^{2} d^{2} e f \,g^{2} x -126 c^{3} d^{3} f^{2} g x +35 a^{3} e^{3} g^{3}-135 a^{2} c d \,e^{2} f \,g^{2}+189 a \,c^{2} d^{2} e \,f^{2} g -105 f^{3} c^{3} d^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (g x +f \right )^{\frac {9}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \sqrt {e x +d}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{\frac {11}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.50, size = 409, normalized size = 1.53 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {x\,\left (-10\,a^3\,c\,d\,e^3\,g^3+54\,a^2\,c^2\,d^2\,e^2\,f\,g^2-126\,a\,c^3\,d^3\,e\,f^2\,g+210\,c^4\,d^4\,f^3\right )}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {70\,a^4\,e^4\,g^3-270\,a^3\,c\,d\,e^3\,f\,g^2+378\,a^2\,c^2\,d^2\,e^2\,f^2\,g-210\,a\,c^3\,d^3\,e\,f^3}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {32\,c^4\,d^4\,x^4}{315\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {4\,c^2\,d^2\,x^2\,\left (a^2\,e^2\,g^2-6\,a\,c\,d\,e\,f\,g+21\,c^2\,d^2\,f^2\right )}{105\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {16\,c^3\,d^3\,x^3\,\left (a\,e\,g-9\,c\,d\,f\right )}{315\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}\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}} \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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