Optimal. Leaf size=412 \[ \frac {16 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{315 c^5 d^5 e g \sqrt {d+e x}}-\frac {16 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right )}{315 c^4 d^4 e}-\frac {4 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right )}{105 c^3 d^3 g \sqrt {d+e x}}-\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (8 a e^2 g+c d (e f-9 d g)\right )}{63 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt {d+e x}} \]
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Rubi [A] time = 0.63, 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 = {880, 870, 794, 648} \[ -\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (8 a e^2 g+c d (e f-9 d g)\right )}{63 c^2 d^2 g \sqrt {d+e x}}-\frac {4 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right )}{105 c^3 d^3 g \sqrt {d+e x}}-\frac {16 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right )}{315 c^4 d^4 e}+\frac {16 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{315 c^5 d^5 e g \sqrt {d+e x}}+\frac {2 e (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt {d+e x}} \]
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)^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 = 0.27, size = 264, normalized size = 0.64 \[ \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (128 a^4 e^5 g^3-16 a^3 c d e^3 g^2 (9 d g+27 e f+4 e g x)+24 a^2 c^2 d^2 e^2 g \left (3 d g (7 f+g x)+e \left (21 f^2+9 f g x+2 g^2 x^2\right )\right )-2 a c^3 d^3 e \left (9 d g \left (35 f^2+14 f g x+3 g^2 x^2\right )+e \left (105 f^3+126 f^2 g x+81 f g^2 x^2+20 g^3 x^3\right )\right )+c^4 d^4 \left (9 d \left (35 f^3+35 f^2 g x+21 f g^2 x^2+5 g^3 x^3\right )+e x \left (105 f^3+189 f^2 g x+135 f g^2 x^2+35 g^3 x^3\right )\right )\right )}{315 c^5 d^5 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 408, normalized size = 0.99 \[ \frac {2 \, {\left (35 \, c^{4} d^{4} e g^{3} x^{4} + 105 \, {\left (3 \, c^{4} d^{5} - 2 \, a c^{3} d^{3} e^{2}\right )} f^{3} - 126 \, {\left (5 \, a c^{3} d^{4} e - 4 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{2} g + 72 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} f g^{2} - 16 \, {\left (9 \, a^{3} c d^{2} e^{3} - 8 \, a^{4} e^{5}\right )} g^{3} + 5 \, {\left (27 \, c^{4} d^{4} e f g^{2} + {\left (9 \, c^{4} d^{5} - 8 \, a c^{3} d^{3} e^{2}\right )} g^{3}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{4} e f^{2} g + 9 \, {\left (7 \, c^{4} d^{5} - 6 \, a c^{3} d^{3} e^{2}\right )} f g^{2} - 2 \, {\left (9 \, a c^{3} d^{4} e - 8 \, a^{2} c^{2} d^{2} e^{3}\right )} g^{3}\right )} x^{2} + {\left (105 \, c^{4} d^{4} e f^{3} + 63 \, {\left (5 \, c^{4} d^{5} - 4 \, a c^{3} d^{3} e^{2}\right )} f^{2} g - 36 \, {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} f g^{2} + 8 \, {\left (9 \, a^{2} c^{2} d^{3} e^{2} - 8 \, a^{3} c d e^{4}\right )} g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{315 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]
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 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{3}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 425, normalized size = 1.03 \[ \frac {2 \left (c d x +a e \right ) \left (35 e \,g^{3} x^{4} c^{4} d^{4}-40 a \,c^{3} d^{3} e^{2} g^{3} x^{3}+45 c^{4} d^{5} g^{3} x^{3}+135 c^{4} d^{4} e f \,g^{2} x^{3}+48 a^{2} c^{2} d^{2} e^{3} g^{3} x^{2}-54 a \,c^{3} d^{4} e \,g^{3} x^{2}-162 a \,c^{3} d^{3} e^{2} f \,g^{2} x^{2}+189 c^{4} d^{5} f \,g^{2} x^{2}+189 c^{4} d^{4} e \,f^{2} g \,x^{2}-64 a^{3} c d \,e^{4} g^{3} x +72 a^{2} c^{2} d^{3} e^{2} g^{3} x +216 a^{2} c^{2} d^{2} e^{3} f \,g^{2} x -252 a \,c^{3} d^{4} e f \,g^{2} x -252 a \,c^{3} d^{3} e^{2} f^{2} g x +315 c^{4} d^{5} f^{2} g x +105 c^{4} d^{4} e \,f^{3} x +128 a^{4} e^{5} g^{3}-144 a^{3} c \,d^{2} e^{3} g^{3}-432 a^{3} c d \,e^{4} f \,g^{2}+504 a^{2} c^{2} d^{3} e^{2} f \,g^{2}+504 a^{2} c^{2} d^{2} e^{3} f^{2} g -630 a \,c^{3} d^{4} e \,f^{2} g -210 a \,c^{3} d^{3} e^{2} f^{3}+315 d^{5} f^{3} c^{4}\right ) \sqrt {e x +d}}{315 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, c^{5} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 484, normalized size = 1.17 \[ \frac {2 \, {\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{3}}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {2 \, {\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} + {\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} - {\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f^{2} g}{5 \, \sqrt {c d x + a e} c^{3} d^{3}} + \frac {2 \, {\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \, {\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} - {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} f g^{2}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} + \frac {2 \, {\left (35 \, c^{5} d^{5} e x^{5} - 144 \, a^{4} c d^{2} e^{4} + 128 \, a^{5} e^{6} + 5 \, {\left (9 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} - {\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \, {\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} - 8 \, {\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} x\right )} g^{3}}{315 \, \sqrt {c d x + a e} c^{5} d^{5}} \]
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 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (256\,a^4\,e^5\,g^3-288\,a^3\,c\,d^2\,e^3\,g^3-864\,a^3\,c\,d\,e^4\,f\,g^2+1008\,a^2\,c^2\,d^3\,e^2\,f\,g^2+1008\,a^2\,c^2\,d^2\,e^3\,f^2\,g-1260\,a\,c^3\,d^4\,e\,f^2\,g-420\,a\,c^3\,d^3\,e^2\,f^3+630\,c^4\,d^5\,f^3\right )}{315\,c^5\,d^5\,e}+\frac {2\,g^3\,x^4\,\sqrt {d+e\,x}}{9\,c\,d}+\frac {x\,\sqrt {d+e\,x}\,\left (-128\,a^3\,c\,d\,e^4\,g^3+144\,a^2\,c^2\,d^3\,e^2\,g^3+432\,a^2\,c^2\,d^2\,e^3\,f\,g^2-504\,a\,c^3\,d^4\,e\,f\,g^2-504\,a\,c^3\,d^3\,e^2\,f^2\,g+630\,c^4\,d^5\,f^2\,g+210\,c^4\,d^4\,e\,f^3\right )}{315\,c^5\,d^5\,e}+\frac {2\,g\,x^2\,\sqrt {d+e\,x}\,\left (16\,a^2\,e^3\,g^2-18\,a\,c\,d^2\,e\,g^2-54\,a\,c\,d\,e^2\,f\,g+63\,c^2\,d^3\,f\,g+63\,c^2\,d^2\,e\,f^2\right )}{105\,c^3\,d^3\,e}+\frac {2\,g^2\,x^3\,\sqrt {d+e\,x}\,\left (9\,c\,g\,d^2+27\,c\,f\,d\,e-8\,a\,g\,e^2\right )}{63\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \]
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 \[ \int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )^{3}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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