Optimal. Leaf size=321 \[ \frac {8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{105 c^4 d^4 e g \sqrt {d+e x}}-\frac {8 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right )}{105 c^3 d^3 e}-\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g+c d (e f-7 d g)\right )}{35 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d g \sqrt {d+e x}} \]
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Rubi [A] time = 0.42, antiderivative size = 321, 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 = {880, 870, 794, 648} \[ -\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g+c d (e f-7 d g)\right )}{35 c^2 d^2 g \sqrt {d+e x}}-\frac {8 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right )}{105 c^3 d^3 e}+\frac {8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{105 c^4 d^4 e g \sqrt {d+e x}}+\frac {2 e (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d g \sqrt {d+e x}} \]
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)^2}{\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)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt {d+e x}}-\frac {1}{7} \left (-7 d+\frac {6 a e^2}{c d}+\frac {e f}{g}\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\\ &=-\frac {2 \left (6 a e^2 g+c d (e f-7 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}}{35 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt {d+e x}}-\frac {\left (4 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 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}{35 c^2 d^2 g}\\ &=-\frac {8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 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}}{105 c^3 d^3 e}-\frac {2 \left (6 a e^2 g+c d (e f-7 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}}{35 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt {d+e x}}+\frac {\left (4 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 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}{105 c^3 d^3 e g}\\ &=\frac {8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 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}}{105 c^4 d^4 e g \sqrt {d+e x}}-\frac {8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 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}}{105 c^3 d^3 e}-\frac {2 \left (6 a e^2 g+c d (e f-7 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}}{35 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 169, normalized size = 0.53 \[ \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (-48 a^3 e^4 g^2+8 a^2 c d e^2 g (7 d g+14 e f+3 e g x)-2 a c^2 d^2 e \left (14 d g (5 f+g x)+e \left (35 f^2+28 f g x+9 g^2 x^2\right )\right )+c^3 d^3 \left (7 d \left (15 f^2+10 f g x+3 g^2 x^2\right )+e x \left (35 f^2+42 f g x+15 g^2 x^2\right )\right )\right )}{105 c^4 d^4 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 256, normalized size = 0.80 \[ \frac {2 \, {\left (15 \, c^{3} d^{3} e g^{2} x^{3} + 35 \, {\left (3 \, c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2}\right )} f^{2} - 28 \, {\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} f g + 8 \, {\left (7 \, a^{2} c d^{2} e^{2} - 6 \, a^{3} e^{4}\right )} g^{2} + 3 \, {\left (14 \, c^{3} d^{3} e f g + {\left (7 \, c^{3} d^{4} - 6 \, a c^{2} d^{2} e^{2}\right )} g^{2}\right )} x^{2} + {\left (35 \, c^{3} d^{3} e f^{2} + 14 \, {\left (5 \, c^{3} d^{4} - 4 \, a c^{2} d^{2} e^{2}\right )} f g - 4 \, {\left (7 \, a c^{2} d^{3} e - 6 \, a^{2} c d e^{3}\right )} g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{105 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]
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 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{2}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 255, normalized size = 0.79 \[ -\frac {2 \left (c d x +a e \right ) \left (-15 e \,g^{2} x^{3} c^{3} d^{3}+18 a \,c^{2} d^{2} e^{2} g^{2} x^{2}-21 c^{3} d^{4} g^{2} x^{2}-42 c^{3} d^{3} e f g \,x^{2}-24 a^{2} c d \,e^{3} g^{2} x +28 a \,c^{2} d^{3} e \,g^{2} x +56 a \,c^{2} d^{2} e^{2} f g x -70 c^{3} d^{4} f g x -35 c^{3} d^{3} e \,f^{2} x +48 a^{3} e^{4} g^{2}-56 a^{2} c \,d^{2} e^{2} g^{2}-112 a^{2} c d \,e^{3} f g +140 a \,c^{2} d^{3} e f g +70 a \,c^{2} d^{2} e^{2} f^{2}-105 d^{4} f^{2} c^{3}\right ) \sqrt {e x +d}}{105 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 309, normalized size = 0.96 \[ \frac {2 \, {\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{2}}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {4 \, {\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} + {\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} - {\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f g}{15 \, \sqrt {c d x + a e} c^{3} d^{3}} + \frac {2 \, {\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \, {\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} - {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} g^{2}}{105 \, \sqrt {c d x + a e} c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.71, size = 279, normalized size = 0.87 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^2\,x^3\,\sqrt {d+e\,x}}{7\,c\,d}-\frac {\sqrt {d+e\,x}\,\left (96\,a^3\,e^4\,g^2-112\,a^2\,c\,d^2\,e^2\,g^2-224\,a^2\,c\,d\,e^3\,f\,g+280\,a\,c^2\,d^3\,e\,f\,g+140\,a\,c^2\,d^2\,e^2\,f^2-210\,c^3\,d^4\,f^2\right )}{105\,c^4\,d^4\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (48\,a^2\,c\,d\,e^3\,g^2-56\,a\,c^2\,d^3\,e\,g^2-112\,a\,c^2\,d^2\,e^2\,f\,g+140\,c^3\,d^4\,f\,g+70\,c^3\,d^3\,e\,f^2\right )}{105\,c^4\,d^4\,e}+\frac {2\,g\,x^2\,\sqrt {d+e\,x}\,\left (7\,c\,g\,d^2+14\,c\,f\,d\,e-6\,a\,g\,e^2\right )}{35\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \]
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 \[ \int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )^{2}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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