Optimal. Leaf size=501 \[ \frac {128 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{3465 c^6 d^6 e g \sqrt {d+e x}}-\frac {128 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right )}{3465 c^5 d^5 e}-\frac {32 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right )}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right )}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (10 a e^2 g+c d (e f-11 d g)\right )}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt {d+e x}} \]
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Rubi [A] time = 0.89, antiderivative size = 501, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {880, 870, 794, 648} \begin {gather*} -\frac {2 (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (10 a e^2 g+c d (e f-11 d g)\right )}{99 c^2 d^2 g \sqrt {d+e x}}-\frac {16 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right )}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {32 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right )}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {128 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right )}{3465 c^5 d^5 e}+\frac {128 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{3465 c^6 d^6 e g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rule 870
Rule 880
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {1}{11} \left (-11 d+\frac {10 a e^2}{c d}+\frac {e f}{g}\right ) \int \frac {\sqrt {d+e x} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {\left (8 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right )\right ) \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{99 c^2 d^2 g}\\ &=-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {\left (16 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right )\right ) \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{231 c^3 d^3 g}\\ &=-\frac {32 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {\left (64 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right )\right ) \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1155 c^4 d^4 g}\\ &=-\frac {128 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3465 c^5 d^5 e}-\frac {32 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}+\frac {\left (64 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3465 c^5 d^5 e g}\\ &=\frac {128 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3465 c^6 d^6 e g \sqrt {d+e x}}-\frac {128 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3465 c^5 d^5 e}-\frac {32 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 246, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (3465 \left (c d^2-a e^2\right ) (c d f-a e g)^4-385 g^3 (a e+c d x)^4 \left (5 a e^2 g-c d (d g+4 e f)\right )+990 g^2 (a e+c d x)^3 (c d f-a e g) \left (c d (2 d g+3 e f)-5 a e^2 g\right )+1386 g (a e+c d x)^2 (c d f-a e g)^2 \left (c d (3 d g+2 e f)-5 a e^2 g\right )+1155 (a e+c d x) (c d f-a e g)^3 \left (c d (4 d g+e f)-5 a e^2 g\right )+315 e g^4 (a e+c d x)^5\right )}{3465 c^6 d^6 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 38.39, size = 8325, normalized size = 16.62 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 597, normalized size = 1.19 \begin {gather*} \frac {2 \, {\left (315 \, c^{5} d^{5} e g^{4} x^{5} + 1155 \, {\left (3 \, c^{5} d^{6} - 2 \, a c^{4} d^{4} e^{2}\right )} f^{4} - 1848 \, {\left (5 \, a c^{4} d^{5} e - 4 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{3} g + 1584 \, {\left (7 \, a^{2} c^{3} d^{4} e^{2} - 6 \, a^{3} c^{2} d^{2} e^{4}\right )} f^{2} g^{2} - 704 \, {\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} f g^{3} + 128 \, {\left (11 \, a^{4} c d^{2} e^{4} - 10 \, a^{5} e^{6}\right )} g^{4} + 35 \, {\left (44 \, c^{5} d^{5} e f g^{3} + {\left (11 \, c^{5} d^{6} - 10 \, a c^{4} d^{4} e^{2}\right )} g^{4}\right )} x^{4} + 10 \, {\left (297 \, c^{5} d^{5} e f^{2} g^{2} + 22 \, {\left (9 \, c^{5} d^{6} - 8 \, a c^{4} d^{4} e^{2}\right )} f g^{3} - 4 \, {\left (11 \, a c^{4} d^{5} e - 10 \, a^{2} c^{3} d^{3} e^{3}\right )} g^{4}\right )} x^{3} + 6 \, {\left (462 \, c^{5} d^{5} e f^{3} g + 99 \, {\left (7 \, c^{5} d^{6} - 6 \, a c^{4} d^{4} e^{2}\right )} f^{2} g^{2} - 44 \, {\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} f g^{3} + 8 \, {\left (11 \, a^{2} c^{3} d^{4} e^{2} - 10 \, a^{3} c^{2} d^{2} e^{4}\right )} g^{4}\right )} x^{2} + {\left (1155 \, c^{5} d^{5} e f^{4} + 924 \, {\left (5 \, c^{5} d^{6} - 4 \, a c^{4} d^{4} e^{2}\right )} f^{3} g - 792 \, {\left (7 \, a c^{4} d^{5} e - 6 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{2} g^{2} + 352 \, {\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} f g^{3} - 64 \, {\left (11 \, a^{3} c^{2} d^{3} e^{3} - 10 \, a^{4} c d e^{5}\right )} g^{4}\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}}{3465 \, {\left (c^{6} d^{6} e x + c^{6} d^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{4}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 641, normalized size = 1.28 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-315 e \,g^{4} x^{5} c^{5} d^{5}+350 a \,c^{4} d^{4} e^{2} g^{4} x^{4}-385 c^{5} d^{6} g^{4} x^{4}-1540 c^{5} d^{5} e f \,g^{3} x^{4}-400 a^{2} c^{3} d^{3} e^{3} g^{4} x^{3}+440 a \,c^{4} d^{5} e \,g^{4} x^{3}+1760 a \,c^{4} d^{4} e^{2} f \,g^{3} x^{3}-1980 c^{5} d^{6} f \,g^{3} x^{3}-2970 c^{5} d^{5} e \,f^{2} g^{2} x^{3}+480 a^{3} c^{2} d^{2} e^{4} g^{4} x^{2}-528 a^{2} c^{3} d^{4} e^{2} g^{4} x^{2}-2112 a^{2} c^{3} d^{3} e^{3} f \,g^{3} x^{2}+2376 a \,c^{4} d^{5} e f \,g^{3} x^{2}+3564 a \,c^{4} d^{4} e^{2} f^{2} g^{2} x^{2}-4158 c^{5} d^{6} f^{2} g^{2} x^{2}-2772 c^{5} d^{5} e \,f^{3} g \,x^{2}-640 a^{4} c d \,e^{5} g^{4} x +704 a^{3} c^{2} d^{3} e^{3} g^{4} x +2816 a^{3} c^{2} d^{2} e^{4} f \,g^{3} x -3168 a^{2} c^{3} d^{4} e^{2} f \,g^{3} x -4752 a^{2} c^{3} d^{3} e^{3} f^{2} g^{2} x +5544 a \,c^{4} d^{5} e \,f^{2} g^{2} x +3696 a \,c^{4} d^{4} e^{2} f^{3} g x -4620 c^{5} d^{6} f^{3} g x -1155 c^{5} d^{5} e \,f^{4} x +1280 a^{5} e^{6} g^{4}-1408 a^{4} c \,d^{2} e^{4} g^{4}-5632 a^{4} c d \,e^{5} f \,g^{3}+6336 a^{3} c^{2} d^{3} e^{3} f \,g^{3}+9504 a^{3} c^{2} d^{2} e^{4} f^{2} g^{2}-11088 a^{2} c^{3} d^{4} e^{2} f^{2} g^{2}-7392 a^{2} c^{3} d^{3} e^{3} f^{3} g +9240 a \,c^{4} d^{5} e \,f^{3} g +2310 a \,c^{4} d^{4} e^{2} f^{4}-3465 d^{6} f^{4} c^{5}\right ) \sqrt {e x +d}}{3465 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, c^{6} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 693, normalized size = 1.38 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{4}}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {8 \, {\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} + {\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} - {\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f^{3} g}{15 \, \sqrt {c d x + a e} c^{3} d^{3}} + \frac {4 \, {\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \, {\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} - {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} f^{2} g^{2}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} + \frac {8 \, {\left (35 \, c^{5} d^{5} e x^{5} - 144 \, a^{4} c d^{2} e^{4} + 128 \, a^{5} e^{6} + 5 \, {\left (9 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} - {\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \, {\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} - 8 \, {\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} x\right )} f g^{3}}{315 \, \sqrt {c d x + a e} c^{5} d^{5}} + \frac {2 \, {\left (315 \, c^{6} d^{6} e x^{6} + 1408 \, a^{5} c d^{2} e^{5} - 1280 \, a^{6} e^{7} + 35 \, {\left (11 \, c^{6} d^{7} - a c^{5} d^{5} e^{2}\right )} x^{5} - 5 \, {\left (11 \, a c^{5} d^{6} e - 10 \, a^{2} c^{4} d^{4} e^{3}\right )} x^{4} + 8 \, {\left (11 \, a^{2} c^{4} d^{5} e^{2} - 10 \, a^{3} c^{3} d^{3} e^{4}\right )} x^{3} - 16 \, {\left (11 \, a^{3} c^{3} d^{4} e^{3} - 10 \, a^{4} c^{2} d^{2} e^{5}\right )} x^{2} + 64 \, {\left (11 \, a^{4} c^{2} d^{3} e^{4} - 10 \, a^{5} c d e^{6}\right )} x\right )} g^{4}}{3465 \, \sqrt {c d x + a e} c^{6} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.09, size = 653, normalized size = 1.30 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^4\,x^5\,\sqrt {d+e\,x}}{11\,c\,d}-\frac {\sqrt {d+e\,x}\,\left (2560\,a^5\,e^6\,g^4-2816\,a^4\,c\,d^2\,e^4\,g^4-11264\,a^4\,c\,d\,e^5\,f\,g^3+12672\,a^3\,c^2\,d^3\,e^3\,f\,g^3+19008\,a^3\,c^2\,d^2\,e^4\,f^2\,g^2-22176\,a^2\,c^3\,d^4\,e^2\,f^2\,g^2-14784\,a^2\,c^3\,d^3\,e^3\,f^3\,g+18480\,a\,c^4\,d^5\,e\,f^3\,g+4620\,a\,c^4\,d^4\,e^2\,f^4-6930\,c^5\,d^6\,f^4\right )}{3465\,c^6\,d^6\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (1280\,a^4\,c\,d\,e^5\,g^4-1408\,a^3\,c^2\,d^3\,e^3\,g^4-5632\,a^3\,c^2\,d^2\,e^4\,f\,g^3+6336\,a^2\,c^3\,d^4\,e^2\,f\,g^3+9504\,a^2\,c^3\,d^3\,e^3\,f^2\,g^2-11088\,a\,c^4\,d^5\,e\,f^2\,g^2-7392\,a\,c^4\,d^4\,e^2\,f^3\,g+9240\,c^5\,d^6\,f^3\,g+2310\,c^5\,d^5\,e\,f^4\right )}{3465\,c^6\,d^6\,e}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-960\,a^3\,c^2\,d^2\,e^4\,g^4+1056\,a^2\,c^3\,d^4\,e^2\,g^4+4224\,a^2\,c^3\,d^3\,e^3\,f\,g^3-4752\,a\,c^4\,d^5\,e\,f\,g^3-7128\,a\,c^4\,d^4\,e^2\,f^2\,g^2+8316\,c^5\,d^6\,f^2\,g^2+5544\,c^5\,d^5\,e\,f^3\,g\right )}{3465\,c^6\,d^6\,e}+\frac {4\,g^2\,x^3\,\sqrt {d+e\,x}\,\left (40\,a^2\,e^3\,g^2-44\,a\,c\,d^2\,e\,g^2-176\,a\,c\,d\,e^2\,f\,g+198\,c^2\,d^3\,f\,g+297\,c^2\,d^2\,e\,f^2\right )}{693\,c^3\,d^3\,e}+\frac {2\,g^3\,x^4\,\sqrt {d+e\,x}\,\left (11\,c\,g\,d^2+44\,c\,f\,d\,e-10\,a\,g\,e^2\right )}{99\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \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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