Optimal. Leaf size=412 \[ \frac {16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{315 c^5 d^5 e g \sqrt {d+e x}}-\frac {16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{315 c^4 d^4 e}-\frac {4 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{63 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt {d+e x}} \]
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Rubi [A]
time = 0.40, antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {894, 884, 808,
662} \begin {gather*} \frac {16 \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 (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{315 c^5 d^5 e g \sqrt {d+e x}}-\frac {16 \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)^2 \left (8 a e^2 g+c d (e f-9 d g)\right )}{315 c^4 d^4 e}-\frac {4 (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) \left (8 a e^2 g+c d (e f-9 d g)\right )}{105 c^3 d^3 g \sqrt {d+e x}}-\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (8 a e^2 g+c d (e f-9 d g)\right )}{63 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 808
Rule 884
Rule 894
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)^3}{\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)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt {d+e x}}-\frac {1}{9} \left (-9 d+\frac {8 a e^2}{c d}+\frac {e f}{g}\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\\ &=-\frac {2 \left (8 a e^2 g+c d (e f-9 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}}{63 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt {d+e x}}-\frac {\left (2 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 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}{21 c^2 d^2 g}\\ &=-\frac {4 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 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}}{105 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (8 a e^2 g+c d (e f-9 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}}{63 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt {d+e x}}-\frac {\left (8 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 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}{105 c^3 d^3 g}\\ &=-\frac {16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 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}}{315 c^4 d^4 e}-\frac {4 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 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}}{105 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (8 a e^2 g+c d (e f-9 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}}{63 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt {d+e x}}+\frac {\left (8 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 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}{315 c^4 d^4 e g}\\ &=\frac {16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 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}}{315 c^5 d^5 e g \sqrt {d+e x}}-\frac {16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 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}}{315 c^4 d^4 e}-\frac {4 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 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}}{105 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (8 a e^2 g+c d (e f-9 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}}{63 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 264, normalized size = 0.64 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} \left (128 a^4 e^5 g^3-16 a^3 c d e^3 g^2 (27 e f+9 d g+4 e g x)+24 a^2 c^2 d^2 e^2 g \left (3 d g (7 f+g x)+e \left (21 f^2+9 f g x+2 g^2 x^2\right )\right )-2 a c^3 d^3 e \left (9 d g \left (35 f^2+14 f g x+3 g^2 x^2\right )+e \left (105 f^3+126 f^2 g x+81 f g^2 x^2+20 g^3 x^3\right )\right )+c^4 d^4 \left (9 d \left (35 f^3+35 f^2 g x+21 f g^2 x^2+5 g^3 x^3\right )+e x \left (105 f^3+189 f^2 g x+135 f g^2 x^2+35 g^3 x^3\right )\right )\right )}{315 c^5 d^5 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 407, normalized size = 0.99
method | result | size |
default | \(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (35 e \,g^{3} x^{4} c^{4} d^{4}-40 a \,c^{3} d^{3} e^{2} g^{3} x^{3}+45 c^{4} d^{5} g^{3} x^{3}+135 c^{4} d^{4} e f \,g^{2} x^{3}+48 a^{2} c^{2} d^{2} e^{3} g^{3} x^{2}-54 a \,c^{3} d^{4} e \,g^{3} x^{2}-162 a \,c^{3} d^{3} e^{2} f \,g^{2} x^{2}+189 c^{4} d^{5} f \,g^{2} x^{2}+189 c^{4} d^{4} e \,f^{2} g \,x^{2}-64 a^{3} c d \,e^{4} g^{3} x +72 a^{2} c^{2} d^{3} e^{2} g^{3} x +216 a^{2} c^{2} d^{2} e^{3} f \,g^{2} x -252 a \,c^{3} d^{4} e f \,g^{2} x -252 a \,c^{3} d^{3} e^{2} f^{2} g x +315 c^{4} d^{5} f^{2} g x +105 c^{4} d^{4} e \,f^{3} x +128 a^{4} e^{5} g^{3}-144 a^{3} c \,d^{2} e^{3} g^{3}-432 a^{3} c d \,e^{4} f \,g^{2}+504 a^{2} c^{2} d^{3} e^{2} f \,g^{2}+504 a^{2} c^{2} d^{2} e^{3} f^{2} g -630 a \,c^{3} d^{4} e \,f^{2} g -210 a \,c^{3} d^{3} e^{2} f^{3}+315 d^{5} f^{3} c^{4}\right )}{315 \sqrt {e x +d}\, c^{5} d^{5}}\) | \(407\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (35 e \,g^{3} x^{4} c^{4} d^{4}-40 a \,c^{3} d^{3} e^{2} g^{3} x^{3}+45 c^{4} d^{5} g^{3} x^{3}+135 c^{4} d^{4} e f \,g^{2} x^{3}+48 a^{2} c^{2} d^{2} e^{3} g^{3} x^{2}-54 a \,c^{3} d^{4} e \,g^{3} x^{2}-162 a \,c^{3} d^{3} e^{2} f \,g^{2} x^{2}+189 c^{4} d^{5} f \,g^{2} x^{2}+189 c^{4} d^{4} e \,f^{2} g \,x^{2}-64 a^{3} c d \,e^{4} g^{3} x +72 a^{2} c^{2} d^{3} e^{2} g^{3} x +216 a^{2} c^{2} d^{2} e^{3} f \,g^{2} x -252 a \,c^{3} d^{4} e f \,g^{2} x -252 a \,c^{3} d^{3} e^{2} f^{2} g x +315 c^{4} d^{5} f^{2} g x +105 c^{4} d^{4} e \,f^{3} x +128 a^{4} e^{5} g^{3}-144 a^{3} c \,d^{2} e^{3} g^{3}-432 a^{3} c d \,e^{4} f \,g^{2}+504 a^{2} c^{2} d^{3} e^{2} f \,g^{2}+504 a^{2} c^{2} d^{2} e^{3} f^{2} g -630 a \,c^{3} d^{4} e \,f^{2} g -210 a \,c^{3} d^{3} e^{2} f^{3}+315 d^{5} f^{3} c^{4}\right ) \sqrt {e x +d}}{315 c^{5} d^{5} \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(425\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 476, normalized size = 1.16 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} e + 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^{3}}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {2 \, {\left (3 \, c^{3} d^{3} x^{3} e - 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^{2} g}{5 \, \sqrt {c d x + a e} c^{3} d^{3}} + \frac {2 \, {\left (15 \, c^{4} d^{4} x^{4} e + 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 g^{2}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} + \frac {2 \, {\left (35 \, c^{5} d^{5} x^{5} e - 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 )} g^{3}}{315 \, \sqrt {c d x + a e} c^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.91, size = 411, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (45 \, c^{4} d^{5} g^{3} x^{3} + 189 \, c^{4} d^{5} f g^{2} x^{2} + 315 \, c^{4} d^{5} f^{2} g x + 315 \, c^{4} d^{5} f^{3} + 128 \, a^{4} g^{3} e^{5} - 16 \, {\left (4 \, a^{3} c d g^{3} x + 27 \, a^{3} c d f g^{2}\right )} e^{4} + 24 \, {\left (2 \, a^{2} c^{2} d^{2} g^{3} x^{2} + 9 \, a^{2} c^{2} d^{2} f g^{2} x + 21 \, a^{2} c^{2} d^{2} f^{2} g - 6 \, a^{3} c d^{2} g^{3}\right )} e^{3} - 2 \, {\left (20 \, a c^{3} d^{3} g^{3} x^{3} + 81 \, a c^{3} d^{3} f g^{2} x^{2} + 105 \, a c^{3} d^{3} f^{3} - 252 \, a^{2} c^{2} d^{3} f g^{2} + 18 \, {\left (7 \, a c^{3} d^{3} f^{2} g - 2 \, a^{2} c^{2} d^{3} g^{3}\right )} x\right )} e^{2} + {\left (35 \, c^{4} d^{4} g^{3} x^{4} + 135 \, c^{4} d^{4} f g^{2} x^{3} - 630 \, a c^{3} d^{4} f^{2} g + 27 \, {\left (7 \, c^{4} d^{4} f^{2} g - 2 \, a c^{3} d^{4} g^{3}\right )} x^{2} + 21 \, {\left (5 \, c^{4} d^{4} f^{3} - 12 \, a c^{3} d^{4} f g^{2}\right )} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{315 \, {\left (c^{5} d^{5} x e + c^{5} d^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )^{3}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1181 vs.
\(2 (396) = 792\).
time = 3.18, size = 1181, normalized size = 2.87 \begin {gather*} \frac {2 \, {\left (c^{4} d^{5} f^{3} - 3 \, a c^{3} d^{4} f^{2} g e - a c^{3} d^{3} f^{3} e^{2} + 3 \, a^{2} c^{2} d^{3} f g^{2} e^{2} + 3 \, a^{2} c^{2} d^{2} f^{2} g e^{3} - a^{3} c d^{2} g^{3} e^{3} - 3 \, a^{3} c d f g^{2} e^{4} + a^{4} g^{3} e^{5}\right )} \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} e^{\left (-1\right )}}{c^{5} d^{5}} + \frac {4 \, {\left (5 \, \sqrt {-c d^{2} e + a e^{3}} c^{4} d^{8} g^{3} - 27 \, \sqrt {-c d^{2} e + a e^{3}} c^{4} d^{7} f g^{2} e + 63 \, \sqrt {-c d^{2} e + a e^{3}} c^{4} d^{6} f^{2} g e^{2} + 7 \, \sqrt {-c d^{2} e + a e^{3}} a c^{3} d^{6} g^{3} e^{2} - 105 \, \sqrt {-c d^{2} e + a e^{3}} c^{4} d^{5} f^{3} e^{3} - 45 \, \sqrt {-c d^{2} e + a e^{3}} a c^{3} d^{5} f g^{2} e^{3} + 189 \, \sqrt {-c d^{2} e + a e^{3}} a c^{3} d^{4} f^{2} g e^{4} + 12 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{2} d^{4} g^{3} e^{4} + 105 \, \sqrt {-c d^{2} e + a e^{3}} a c^{3} d^{3} f^{3} e^{5} - 144 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{2} d^{3} f g^{2} e^{5} - 252 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{2} d^{2} f^{2} g e^{6} + 40 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c d^{2} g^{3} e^{6} + 216 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c d f g^{2} e^{7} - 64 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} g^{3} e^{8}\right )} e^{\left (-4\right )}}{315 \, c^{5} d^{5}} + \frac {2 \, {\left (315 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{4} f^{2} g e^{5} + 105 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{3} f^{3} e^{6} - 630 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{2} d^{3} f g^{2} e^{6} + 189 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{2} d^{3} f g^{2} e^{3} - 630 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{2} d^{2} f^{2} g e^{7} + 315 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c d^{2} g^{3} e^{7} + 189 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{2} d^{2} f^{2} g e^{4} - 189 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c d^{2} g^{3} e^{4} + 45 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c d^{2} g^{3} e + 945 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c d f g^{2} e^{8} - 567 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c d f g^{2} e^{5} + 135 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c d f g^{2} e^{2} - 420 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} g^{3} e^{9} + 378 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} g^{3} e^{6} - 180 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a g^{3} e^{3} + 35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}} g^{3}\right )} e^{\left (-8\right )}}{315 \, c^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.86, size = 438, normalized size = 1.06 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (256\,a^4\,e^5\,g^3-288\,a^3\,c\,d^2\,e^3\,g^3-864\,a^3\,c\,d\,e^4\,f\,g^2+1008\,a^2\,c^2\,d^3\,e^2\,f\,g^2+1008\,a^2\,c^2\,d^2\,e^3\,f^2\,g-1260\,a\,c^3\,d^4\,e\,f^2\,g-420\,a\,c^3\,d^3\,e^2\,f^3+630\,c^4\,d^5\,f^3\right )}{315\,c^5\,d^5\,e}+\frac {2\,g^3\,x^4\,\sqrt {d+e\,x}}{9\,c\,d}+\frac {x\,\sqrt {d+e\,x}\,\left (-128\,a^3\,c\,d\,e^4\,g^3+144\,a^2\,c^2\,d^3\,e^2\,g^3+432\,a^2\,c^2\,d^2\,e^3\,f\,g^2-504\,a\,c^3\,d^4\,e\,f\,g^2-504\,a\,c^3\,d^3\,e^2\,f^2\,g+630\,c^4\,d^5\,f^2\,g+210\,c^4\,d^4\,e\,f^3\right )}{315\,c^5\,d^5\,e}+\frac {2\,g\,x^2\,\sqrt {d+e\,x}\,\left (16\,a^2\,e^3\,g^2-18\,a\,c\,d^2\,e\,g^2-54\,a\,c\,d\,e^2\,f\,g+63\,c^2\,d^3\,f\,g+63\,c^2\,d^2\,e\,f^2\right )}{105\,c^3\,d^3\,e}+\frac {2\,g^2\,x^3\,\sqrt {d+e\,x}\,\left (9\,c\,g\,d^2+27\,c\,f\,d\,e-8\,a\,g\,e^2\right )}{63\,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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