Optimal. Leaf size=257 \[ -\frac {16 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (2 a e^2 g-c d (3 e f-d g)\right )}{5 c^4 d^4 e \sqrt {d+e x}}+\frac {16 g^2 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{5 c^3 d^3 e}+\frac {12 g (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c^2 d^2 \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} (f+g x)^3}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.33, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {866, 870, 794, 648} \[ \frac {16 g^2 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{5 c^3 d^3 e}+\frac {12 g (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c^2 d^2 \sqrt {d+e x}}-\frac {16 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (2 a e^2 g-c d (3 e f-d g)\right )}{5 c^4 d^4 e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} (f+g x)^3}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rule 866
Rule 870
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 \sqrt {d+e x} (f+g x)^3}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(6 g) \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}{c d}\\ &=-\frac {2 \sqrt {d+e x} (f+g x)^3}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {12 g (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^2 d^2 \sqrt {d+e x}}+\frac {(24 g (c d f-a e g)) \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}{5 c^2 d^2}\\ &=-\frac {2 \sqrt {d+e x} (f+g x)^3}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 g^2 (c d f-a e g) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^3 d^3 e}+\frac {12 g (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^2 d^2 \sqrt {d+e x}}-\frac {\left (8 g (c d f-a e g) \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}{5 c^3 d^3 e}\\ &=-\frac {2 \sqrt {d+e x} (f+g x)^3}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {16 g (c d f-a e g) \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}}{5 c^4 d^4 e \sqrt {d+e x}}+\frac {16 g^2 (c d f-a e g) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^3 d^3 e}+\frac {12 g (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^2 d^2 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 134, normalized size = 0.52 \[ \frac {2 \sqrt {d+e x} \left (16 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (g x-5 f)-2 a c^2 d^2 e g \left (-15 f^2+10 f g x+g^2 x^2\right )+c^3 d^3 \left (-5 f^3+15 f^2 g x+5 f g^2 x^2+g^3 x^3\right )\right )}{5 c^4 d^4 \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 216, normalized size = 0.84 \[ \frac {2 \, {\left (c^{3} d^{3} g^{3} x^{3} - 5 \, c^{3} d^{3} f^{3} + 30 \, a c^{2} d^{2} e f^{2} g - 40 \, a^{2} c d e^{2} f g^{2} + 16 \, a^{3} e^{3} g^{3} + {\left (5 \, c^{3} d^{3} f g^{2} - 2 \, a c^{2} d^{2} e g^{3}\right )} x^{2} + {\left (15 \, c^{3} d^{3} f^{2} g - 20 \, a c^{2} d^{2} e f g^{2} + 8 \, a^{2} c d e^{2} g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{5 \, {\left (c^{5} d^{5} e x^{2} + a c^{4} d^{5} e + {\left (c^{5} d^{6} + a c^{4} d^{4} e^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 187, normalized size = 0.73 \[ \frac {2 \left (c d x +a e \right ) \left (g^{3} x^{3} c^{3} d^{3}-2 a \,c^{2} d^{2} e \,g^{3} x^{2}+5 c^{3} d^{3} f \,g^{2} x^{2}+8 a^{2} c d \,e^{2} g^{3} x -20 a \,c^{2} d^{2} e f \,g^{2} x +15 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-40 a^{2} c d \,e^{2} f \,g^{2}+30 a \,c^{2} d^{2} e \,f^{2} g -5 f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{5 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}} c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 165, normalized size = 0.64 \[ -\frac {2 \, f^{3}}{\sqrt {c d x + a e} c d} + \frac {6 \, {\left (c d x + 2 \, a e\right )} f^{2} g}{\sqrt {c d x + a e} c^{2} d^{2}} + \frac {2 \, {\left (c^{2} d^{2} x^{2} - 4 \, a c d e x - 8 \, a^{2} e^{2}\right )} f g^{2}}{\sqrt {c d x + a e} c^{3} d^{3}} + \frac {2 \, {\left (c^{3} d^{3} x^{3} - 2 \, a c^{2} d^{2} e x^{2} + 8 \, a^{2} c d e^{2} x + 16 \, a^{3} e^{3}\right )} g^{3}}{5 \, \sqrt {c d x + a e} c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.61, size = 252, normalized size = 0.98 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (32\,a^3\,e^3\,g^3-80\,a^2\,c\,d\,e^2\,f\,g^2+60\,a\,c^2\,d^2\,e\,f^2\,g-10\,c^3\,d^3\,f^3\right )}{5\,c^5\,d^5\,e}+\frac {2\,g^3\,x^3\,\sqrt {d+e\,x}}{5\,c^2\,d^2\,e}-\frac {2\,g^2\,x^2\,\left (2\,a\,e\,g-5\,c\,d\,f\right )\,\sqrt {d+e\,x}}{5\,c^3\,d^3\,e}+\frac {2\,g\,x\,\sqrt {d+e\,x}\,\left (8\,a^2\,e^2\,g^2-20\,a\,c\,d\,e\,f\,g+15\,c^2\,d^2\,f^2\right )}{5\,c^4\,d^4\,e}\right )}{\frac {a}{c}+x^2+\frac {x\,\left (5\,c^5\,d^6+5\,a\,c^4\,d^4\,e^2\right )}{5\,c^5\,d^5\,e}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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