Optimal. Leaf size=267 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{9/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 (c d f-a e g)^4 (d+e x)^{5/2} (f+g x)^{5/2}} \]
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Rubi [A]
time = 0.22, 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 = {886, 874}
\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 )^{5/2}}{1155 (d+e x)^{5/2} (f+g x)^{5/2} (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 )^{5/2}}{231 (d+e x)^{5/2} (f+g x)^{7/2} (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 )^{5/2}}{33 (d+e x)^{5/2} (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 )^{5/2}}{11 (d+e x)^{5/2} (f+g x)^{11/2} (c d f-a e g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 874
Rule 886
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)^{13/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {(6 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)^{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 )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (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 )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx}{33 (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}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (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 )^{5/2}}{231 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {\left (16 c^3 d^3\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}{231 (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 )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (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 )^{5/2}}{231 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{7/2}}+\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 )^{5/2}}{1155 (c d f-a e g)^4 (d+e x)^{5/2} (f+g x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 141, normalized size = 0.53 \begin {gather*} \frac {2 (a e+c d x)^4 ((a e+c d x) (d+e x))^{3/2} \left (-105 g^3+\frac {385 c d g^2 (f+g x)}{a e+c d x}-\frac {495 c^2 d^2 g (f+g x)^2}{(a e+c d x)^2}+\frac {231 c^3 d^3 (f+g x)^3}{(a e+c d x)^3}\right )}{1155 (c d f-a e g)^4 (d+e x)^{3/2} (f+g x)^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 267, normalized size = 1.00
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+40 a \,c^{2} d^{2} e \,g^{3} x^{2}-88 c^{3} d^{3} f \,g^{2} x^{2}-70 a^{2} c d \,e^{2} g^{3} x +220 a \,c^{2} d^{2} e f \,g^{2} x -198 c^{3} d^{3} f^{2} g x +105 a^{3} e^{3} g^{3}-385 a^{2} c d \,e^{2} f \,g^{2}+495 a \,c^{2} d^{2} e \,f^{2} g -231 f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{1155 \left (g x +f \right )^{\frac {11}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}\) | \(260\) |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-16 g^{3} x^{4} c^{4} d^{4}+24 a \,c^{3} d^{3} e \,g^{3} x^{3}-88 c^{4} d^{4} f \,g^{2} x^{3}-30 a^{2} c^{2} d^{2} e^{2} g^{3} x^{2}+132 a \,c^{3} d^{3} e f \,g^{2} x^{2}-198 c^{4} d^{4} f^{2} g \,x^{2}+35 a^{3} c d \,e^{3} g^{3} x -165 a^{2} c^{2} d^{2} e^{2} f \,g^{2} x +297 a \,c^{3} d^{3} e \,f^{2} g x -231 c^{4} d^{4} f^{3} x +105 g^{3} e^{4} a^{4}-385 a^{3} c d \,e^{3} f \,g^{2}+495 a^{2} c^{2} d^{2} e^{2} f^{2} g -231 a \,c^{3} d^{3} e \,f^{3}\right ) \left (c d x +a e \right )}{1155 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {11}{2}} \left (a e g -c d f \right )^{4}}\) | \(267\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1496 vs.
\(2 (247) = 494\).
time = 1.01, size = 1496, normalized size = 5.60 \begin {gather*} \frac {2 \, {\left (16 \, c^{5} d^{5} g^{3} x^{5} + 88 \, c^{5} d^{5} f g^{2} x^{4} + 198 \, c^{5} d^{5} f^{2} g x^{3} + 231 \, c^{5} d^{5} f^{3} x^{2} - 105 \, a^{5} g^{3} e^{5} - 35 \, {\left (4 \, a^{4} c d g^{3} x - 11 \, a^{4} c d f g^{2}\right )} e^{4} - 5 \, {\left (a^{3} c^{2} d^{2} g^{3} x^{2} - 110 \, a^{3} c^{2} d^{2} f g^{2} x + 99 \, a^{3} c^{2} d^{2} f^{2} g\right )} e^{3} + 3 \, {\left (2 \, a^{2} c^{3} d^{3} g^{3} x^{3} + 11 \, a^{2} c^{3} d^{3} f g^{2} x^{2} - 264 \, a^{2} c^{3} d^{3} f^{2} g x + 77 \, a^{2} c^{3} d^{3} f^{3}\right )} e^{2} - {\left (8 \, a c^{4} d^{4} g^{3} x^{4} + 44 \, a c^{4} d^{4} f g^{2} x^{3} + 99 \, a c^{4} d^{4} f^{2} g x^{2} - 462 \, a c^{4} d^{4} f^{3} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d}}{1155 \, {\left (c^{4} d^{5} f^{4} g^{6} x^{6} + 6 \, c^{4} d^{5} f^{5} g^{5} x^{5} + 15 \, c^{4} d^{5} f^{6} g^{4} x^{4} + 20 \, c^{4} d^{5} f^{7} g^{3} x^{3} + 15 \, c^{4} d^{5} f^{8} g^{2} x^{2} + 6 \, c^{4} d^{5} f^{9} g x + c^{4} d^{5} f^{10} + {\left (a^{4} g^{10} x^{7} + 6 \, a^{4} f g^{9} x^{6} + 15 \, a^{4} f^{2} g^{8} x^{5} + 20 \, a^{4} f^{3} g^{7} x^{4} + 15 \, a^{4} f^{4} g^{6} x^{3} + 6 \, a^{4} f^{5} g^{5} x^{2} + a^{4} f^{6} g^{4} x\right )} e^{5} - {\left (4 \, a^{3} c d f g^{9} x^{7} - a^{4} d f^{6} g^{4} + {\left (24 \, a^{3} c d f^{2} g^{8} - a^{4} d g^{10}\right )} x^{6} + 6 \, {\left (10 \, a^{3} c d f^{3} g^{7} - a^{4} d f g^{9}\right )} x^{5} + 5 \, {\left (16 \, a^{3} c d f^{4} g^{6} - 3 \, a^{4} d f^{2} g^{8}\right )} x^{4} + 20 \, {\left (3 \, a^{3} c d f^{5} g^{5} - a^{4} d f^{3} g^{7}\right )} x^{3} + 3 \, {\left (8 \, a^{3} c d f^{6} g^{4} - 5 \, a^{4} d f^{4} g^{6}\right )} x^{2} + 2 \, {\left (2 \, a^{3} c d f^{7} g^{3} - 3 \, a^{4} d f^{5} g^{5}\right )} x\right )} e^{4} + 2 \, {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{8} x^{7} - 2 \, a^{3} c d^{2} f^{7} g^{3} + 2 \, {\left (9 \, a^{2} c^{2} d^{2} f^{3} g^{7} - a^{3} c d^{2} f g^{9}\right )} x^{6} + 3 \, {\left (15 \, a^{2} c^{2} d^{2} f^{4} g^{6} - 4 \, a^{3} c d^{2} f^{2} g^{8}\right )} x^{5} + 30 \, {\left (2 \, a^{2} c^{2} d^{2} f^{5} g^{5} - a^{3} c d^{2} f^{3} g^{7}\right )} x^{4} + 5 \, {\left (9 \, a^{2} c^{2} d^{2} f^{6} g^{4} - 8 \, a^{3} c d^{2} f^{4} g^{6}\right )} x^{3} + 6 \, {\left (3 \, a^{2} c^{2} d^{2} f^{7} g^{3} - 5 \, a^{3} c d^{2} f^{5} g^{5}\right )} x^{2} + 3 \, {\left (a^{2} c^{2} d^{2} f^{8} g^{2} - 4 \, a^{3} c d^{2} f^{6} g^{4}\right )} x\right )} e^{3} - 2 \, {\left (2 \, a c^{3} d^{3} f^{3} g^{7} x^{7} - 3 \, a^{2} c^{2} d^{3} f^{8} g^{2} + 3 \, {\left (4 \, a c^{3} d^{3} f^{4} g^{6} - a^{2} c^{2} d^{3} f^{2} g^{8}\right )} x^{6} + 6 \, {\left (5 \, a c^{3} d^{3} f^{5} g^{5} - 3 \, a^{2} c^{2} d^{3} f^{3} g^{7}\right )} x^{5} + 5 \, {\left (8 \, a c^{3} d^{3} f^{6} g^{4} - 9 \, a^{2} c^{2} d^{3} f^{4} g^{6}\right )} x^{4} + 30 \, {\left (a c^{3} d^{3} f^{7} g^{3} - 2 \, a^{2} c^{2} d^{3} f^{5} g^{5}\right )} x^{3} + 3 \, {\left (4 \, a c^{3} d^{3} f^{8} g^{2} - 15 \, a^{2} c^{2} d^{3} f^{6} g^{4}\right )} x^{2} + 2 \, {\left (a c^{3} d^{3} f^{9} g - 9 \, a^{2} c^{2} d^{3} f^{7} g^{3}\right )} x\right )} e^{2} + {\left (c^{4} d^{4} f^{4} g^{6} x^{7} - 4 \, a c^{3} d^{4} f^{9} g + 2 \, {\left (3 \, c^{4} d^{4} f^{5} g^{5} - 2 \, a c^{3} d^{4} f^{3} g^{7}\right )} x^{6} + 3 \, {\left (5 \, c^{4} d^{4} f^{6} g^{4} - 8 \, a c^{3} d^{4} f^{4} g^{6}\right )} x^{5} + 20 \, {\left (c^{4} d^{4} f^{7} g^{3} - 3 \, a c^{3} d^{4} f^{5} g^{5}\right )} x^{4} + 5 \, {\left (3 \, c^{4} d^{4} f^{8} g^{2} - 16 \, a c^{3} d^{4} f^{6} g^{4}\right )} x^{3} + 6 \, {\left (c^{4} d^{4} f^{9} g - 10 \, a c^{3} d^{4} f^{7} g^{3}\right )} x^{2} + {\left (c^{4} d^{4} f^{10} - 24 \, a c^{3} d^{4} f^{8} g^{2}\right )} x\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F(-1)] Timed out
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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Mupad [B]
time = 4.83, size = 519, normalized size = 1.94 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {210\,a^5\,e^5\,g^3-770\,a^4\,c\,d\,e^4\,f\,g^2+990\,a^3\,c^2\,d^2\,e^3\,f^2\,g-462\,a^2\,c^3\,d^3\,e^2\,f^3}{1155\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {x^2\,\left (-10\,a^3\,c^2\,d^2\,e^3\,g^3+66\,a^2\,c^3\,d^3\,e^2\,f\,g^2-198\,a\,c^4\,d^4\,e\,f^2\,g+462\,c^5\,d^5\,f^3\right )}{1155\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {32\,c^5\,d^5\,x^5}{1155\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {4\,c^3\,d^3\,x^3\,\left (3\,a^2\,e^2\,g^2-22\,a\,c\,d\,e\,f\,g+99\,c^2\,d^2\,f^2\right )}{1155\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c^4\,d^4\,x^4\,\left (a\,e\,g-11\,c\,d\,f\right )}{1155\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {4\,a\,c\,d\,e\,x\,\left (70\,a^3\,e^3\,g^3-275\,a^2\,c\,d\,e^2\,f\,g^2+396\,a\,c^2\,d^2\,e\,f^2\,g-231\,c^3\,d^3\,f^3\right )}{1155\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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