Optimal. Leaf size=267 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 (c d f-a e g)^4 (d+e x)^{7/2} (f+g x)^{7/2}} \]
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Rubi [A]
time = 0.21, 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 )^{7/2}}{3003 (d+e x)^{7/2} (f+g x)^{7/2} (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 )^{7/2}}{429 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)^3}+\frac {12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^{7/2} (f+g x)^{11/2} (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}}{13 (d+e x)^{7/2} (f+g x)^{13/2} (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 )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\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 )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx}{13 (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}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {\left (24 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)^{11/2}} \, dx}{143 (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}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\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}}{429 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{9/2}}+\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 )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx}{429 (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 )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\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}}{429 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{9/2}}+\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 )^{7/2}}{3003 (c d f-a e g)^4 (d+e x)^{7/2} (f+g x)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 141, normalized size = 0.53 \begin {gather*} \frac {2 (a e+c d x)^4 ((a e+c d x) (d+e x))^{5/2} \left (-231 g^3+\frac {819 c d g^2 (f+g x)}{a e+c d x}-\frac {1001 c^2 d^2 g (f+g x)^2}{(a e+c d x)^2}+\frac {429 c^3 d^3 (f+g x)^3}{(a e+c d x)^3}\right )}{3003 (c d f-a e g)^4 (d+e x)^{5/2} (f+g x)^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 349, normalized size = 1.31
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+56 a \,c^{2} d^{2} e \,g^{3} x^{2}-104 c^{3} d^{3} f \,g^{2} x^{2}-126 a^{2} c d \,e^{2} g^{3} x +364 a \,c^{2} d^{2} e f \,g^{2} x -286 c^{3} d^{3} f^{2} g x +231 a^{3} e^{3} g^{3}-819 a^{2} c d \,e^{2} f \,g^{2}+1001 a \,c^{2} d^{2} e \,f^{2} g -429 f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 \left (g x +f \right )^{\frac {13}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}}}\) | \(260\) |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-16 c^{5} d^{5} g^{3} x^{5}+24 a \,c^{4} d^{4} e \,g^{3} x^{4}-104 c^{5} d^{5} f \,g^{2} x^{4}-30 a^{2} c^{3} d^{3} e^{2} g^{3} x^{3}+156 a \,c^{4} d^{4} e f \,g^{2} x^{3}-286 c^{5} d^{5} f^{2} g \,x^{3}+35 a^{3} c^{2} d^{2} e^{3} g^{3} x^{2}-195 a^{2} c^{3} d^{3} e^{2} f \,g^{2} x^{2}+429 a \,c^{4} d^{4} e \,f^{2} g \,x^{2}-429 c^{5} d^{5} f^{3} x^{2}+336 a^{4} c d \,e^{4} g^{3} x -1274 a^{3} c^{2} d^{2} e^{3} f \,g^{2} x +1716 a^{2} c^{3} d^{3} e^{2} f^{2} g x -858 a \,c^{4} d^{4} e \,f^{3} x +231 a^{5} e^{5} g^{3}-819 a^{4} c d \,e^{4} f \,g^{2}+1001 a^{3} c^{2} d^{2} e^{3} f^{2} g -429 a^{2} c^{3} d^{3} e^{2} f^{3}\right ) \left (c d x +a e \right )}{3003 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {13}{2}} \left (a e g -c d f \right )^{4}}\) | \(349\) |
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. 1741 vs.
\(2 (247) = 494\).
time = 0.95, size = 1741, normalized size = 6.52 \begin {gather*} \frac {2 \, {\left (16 \, c^{6} d^{6} g^{3} x^{6} + 104 \, c^{6} d^{6} f g^{2} x^{5} + 286 \, c^{6} d^{6} f^{2} g x^{4} + 429 \, c^{6} d^{6} f^{3} x^{3} - 231 \, a^{6} g^{3} e^{6} - 63 \, {\left (9 \, a^{5} c d g^{3} x - 13 \, a^{5} c d f g^{2}\right )} e^{5} - 7 \, {\left (53 \, a^{4} c^{2} d^{2} g^{3} x^{2} - 299 \, a^{4} c^{2} d^{2} f g^{2} x + 143 \, a^{4} c^{2} d^{2} f^{2} g\right )} e^{4} - {\left (5 \, a^{3} c^{3} d^{3} g^{3} x^{3} - 1469 \, a^{3} c^{3} d^{3} f g^{2} x^{2} + 2717 \, a^{3} c^{3} d^{3} f^{2} g x - 429 \, a^{3} c^{3} d^{3} f^{3}\right )} e^{3} + 3 \, {\left (2 \, a^{2} c^{4} d^{4} g^{3} x^{4} + 13 \, a^{2} c^{4} d^{4} f g^{2} x^{3} - 715 \, a^{2} c^{4} d^{4} f^{2} g x^{2} + 429 \, a^{2} c^{4} d^{4} f^{3} x\right )} e^{2} - {\left (8 \, a c^{5} d^{5} g^{3} x^{5} + 52 \, a c^{5} d^{5} f g^{2} x^{4} + 143 \, a c^{5} d^{5} f^{2} g x^{3} - 1287 \, a c^{5} d^{5} f^{3} x^{2}\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}}{3003 \, {\left (c^{4} d^{5} f^{4} g^{7} x^{7} + 7 \, c^{4} d^{5} f^{5} g^{6} x^{6} + 21 \, c^{4} d^{5} f^{6} g^{5} x^{5} + 35 \, c^{4} d^{5} f^{7} g^{4} x^{4} + 35 \, c^{4} d^{5} f^{8} g^{3} x^{3} + 21 \, c^{4} d^{5} f^{9} g^{2} x^{2} + 7 \, c^{4} d^{5} f^{10} g x + c^{4} d^{5} f^{11} + {\left (a^{4} g^{11} x^{8} + 7 \, a^{4} f g^{10} x^{7} + 21 \, a^{4} f^{2} g^{9} x^{6} + 35 \, a^{4} f^{3} g^{8} x^{5} + 35 \, a^{4} f^{4} g^{7} x^{4} + 21 \, a^{4} f^{5} g^{6} x^{3} + 7 \, a^{4} f^{6} g^{5} x^{2} + a^{4} f^{7} g^{4} x\right )} e^{5} - {\left (4 \, a^{3} c d f g^{10} x^{8} - a^{4} d f^{7} g^{4} + {\left (28 \, a^{3} c d f^{2} g^{9} - a^{4} d g^{11}\right )} x^{7} + 7 \, {\left (12 \, a^{3} c d f^{3} g^{8} - a^{4} d f g^{10}\right )} x^{6} + 7 \, {\left (20 \, a^{3} c d f^{4} g^{7} - 3 \, a^{4} d f^{2} g^{9}\right )} x^{5} + 35 \, {\left (4 \, a^{3} c d f^{5} g^{6} - a^{4} d f^{3} g^{8}\right )} x^{4} + 7 \, {\left (12 \, a^{3} c d f^{6} g^{5} - 5 \, a^{4} d f^{4} g^{7}\right )} x^{3} + 7 \, {\left (4 \, a^{3} c d f^{7} g^{4} - 3 \, a^{4} d f^{5} g^{6}\right )} x^{2} + {\left (4 \, a^{3} c d f^{8} g^{3} - 7 \, a^{4} d f^{6} g^{5}\right )} x\right )} e^{4} + 2 \, {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{9} x^{8} - 2 \, a^{3} c d^{2} f^{8} g^{3} + {\left (21 \, a^{2} c^{2} d^{2} f^{3} g^{8} - 2 \, a^{3} c d^{2} f g^{10}\right )} x^{7} + 7 \, {\left (9 \, a^{2} c^{2} d^{2} f^{4} g^{7} - 2 \, a^{3} c d^{2} f^{2} g^{9}\right )} x^{6} + 21 \, {\left (5 \, a^{2} c^{2} d^{2} f^{5} g^{6} - 2 \, a^{3} c d^{2} f^{3} g^{8}\right )} x^{5} + 35 \, {\left (3 \, a^{2} c^{2} d^{2} f^{6} g^{5} - 2 \, a^{3} c d^{2} f^{4} g^{7}\right )} x^{4} + 7 \, {\left (9 \, a^{2} c^{2} d^{2} f^{7} g^{4} - 10 \, a^{3} c d^{2} f^{5} g^{6}\right )} x^{3} + 21 \, {\left (a^{2} c^{2} d^{2} f^{8} g^{3} - 2 \, a^{3} c d^{2} f^{6} g^{5}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{2} f^{9} g^{2} - 14 \, a^{3} c d^{2} f^{7} g^{4}\right )} x\right )} e^{3} - 2 \, {\left (2 \, a c^{3} d^{3} f^{3} g^{8} x^{8} - 3 \, a^{2} c^{2} d^{3} f^{9} g^{2} + {\left (14 \, a c^{3} d^{3} f^{4} g^{7} - 3 \, a^{2} c^{2} d^{3} f^{2} g^{9}\right )} x^{7} + 21 \, {\left (2 \, a c^{3} d^{3} f^{5} g^{6} - a^{2} c^{2} d^{3} f^{3} g^{8}\right )} x^{6} + 7 \, {\left (10 \, a c^{3} d^{3} f^{6} g^{5} - 9 \, a^{2} c^{2} d^{3} f^{4} g^{7}\right )} x^{5} + 35 \, {\left (2 \, a c^{3} d^{3} f^{7} g^{4} - 3 \, a^{2} c^{2} d^{3} f^{5} g^{6}\right )} x^{4} + 21 \, {\left (2 \, a c^{3} d^{3} f^{8} g^{3} - 5 \, a^{2} c^{2} d^{3} f^{6} g^{5}\right )} x^{3} + 7 \, {\left (2 \, a c^{3} d^{3} f^{9} g^{2} - 9 \, a^{2} c^{2} d^{3} f^{7} g^{4}\right )} x^{2} + {\left (2 \, a c^{3} d^{3} f^{10} g - 21 \, a^{2} c^{2} d^{3} f^{8} g^{3}\right )} x\right )} e^{2} + {\left (c^{4} d^{4} f^{4} g^{7} x^{8} - 4 \, a c^{3} d^{4} f^{10} g + {\left (7 \, c^{4} d^{4} f^{5} g^{6} - 4 \, a c^{3} d^{4} f^{3} g^{8}\right )} x^{7} + 7 \, {\left (3 \, c^{4} d^{4} f^{6} g^{5} - 4 \, a c^{3} d^{4} f^{4} g^{7}\right )} x^{6} + 7 \, {\left (5 \, c^{4} d^{4} f^{7} g^{4} - 12 \, a c^{3} d^{4} f^{5} g^{6}\right )} x^{5} + 35 \, {\left (c^{4} d^{4} f^{8} g^{3} - 4 \, a c^{3} d^{4} f^{6} g^{5}\right )} x^{4} + 7 \, {\left (3 \, c^{4} d^{4} f^{9} g^{2} - 20 \, a c^{3} d^{4} f^{7} g^{4}\right )} x^{3} + 7 \, {\left (c^{4} d^{4} f^{10} g - 12 \, a c^{3} d^{4} f^{8} g^{3}\right )} x^{2} + {\left (c^{4} d^{4} f^{11} - 28 \, a c^{3} d^{4} f^{9} 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 = 5.12, size = 627, normalized size = 2.35 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {462\,a^6\,e^6\,g^3-1638\,a^5\,c\,d\,e^5\,f\,g^2+2002\,a^4\,c^2\,d^2\,e^4\,f^2\,g-858\,a^3\,c^3\,d^3\,e^3\,f^3}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {x^3\,\left (-10\,a^3\,c^3\,d^3\,e^3\,g^3+78\,a^2\,c^4\,d^4\,e^2\,f\,g^2-286\,a\,c^5\,d^5\,e\,f^2\,g+858\,c^6\,d^6\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {32\,c^6\,d^6\,x^6}{3003\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {4\,c^4\,d^4\,x^4\,\left (3\,a^2\,e^2\,g^2-26\,a\,c\,d\,e\,f\,g+143\,c^2\,d^2\,f^2\right )}{3003\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c^5\,d^5\,x^5\,\left (a\,e\,g-13\,c\,d\,f\right )}{3003\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {2\,a^2\,c\,d\,e^2\,x\,\left (567\,a^3\,e^3\,g^3-2093\,a^2\,c\,d\,e^2\,f\,g^2+2717\,a\,c^2\,d^2\,e\,f^2\,g-1287\,c^3\,d^3\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {2\,a\,c^2\,d^2\,e\,x^2\,\left (371\,a^3\,e^3\,g^3-1469\,a^2\,c\,d\,e^2\,f\,g^2+2145\,a\,c^2\,d^2\,e\,f^2\,g-1287\,c^3\,d^3\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^6\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^6\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^6}+\frac {6\,f\,x^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {6\,f^5\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^5}+\frac {15\,f^2\,x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {20\,f^3\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {15\,f^4\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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