Optimal. Leaf size=181 \[ -\frac {8 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^3 d^3 e \sqrt {d+e x}}+\frac {8 g^2 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 e}-\frac {2 \sqrt {d+e x} (f+g x)^2}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.18, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {866, 794, 648} \[ -\frac {8 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^3 d^3 e \sqrt {d+e x}}+\frac {8 g^2 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 e}-\frac {2 \sqrt {d+e x} (f+g x)^2}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rule 866
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)^2}{\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)^2}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(4 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}{c d}\\ &=-\frac {2 \sqrt {d+e x} (f+g x)^2}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {8 g^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e}-\frac {\left (4 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}{3 c^2 d^2 e}\\ &=-\frac {2 \sqrt {d+e x} (f+g x)^2}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 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}}{3 c^3 d^3 e \sqrt {d+e x}}+\frac {8 g^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 88, normalized size = 0.49 \[ \frac {2 \sqrt {d+e x} \left (-8 a^2 e^2 g^2-4 a c d e g (g x-3 f)+c^2 d^2 \left (-3 f^2+6 f g x+g^2 x^2\right )\right )}{3 c^3 d^3 \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 147, normalized size = 0.81 \[ \frac {2 \, {\left (c^{2} d^{2} g^{2} x^{2} - 3 \, c^{2} d^{2} f^{2} + 12 \, a c d e f g - 8 \, a^{2} e^{2} g^{2} + 2 \, {\left (3 \, c^{2} d^{2} f g - 2 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3 \, {\left (c^{4} d^{4} e x^{2} + a c^{3} d^{4} e + {\left (c^{4} d^{5} + a c^{3} d^{3} e^{2}\right )} x\right )}} \]
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 = 116, normalized size = 0.64 \[ -\frac {2 \left (c d x +a e \right ) \left (-g^{2} x^{2} c^{2} d^{2}+4 a c d e \,g^{2} x -6 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-12 a c d e f g +3 f^{2} c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}} c^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 98, normalized size = 0.54 \[ -\frac {2 \, f^{2}}{\sqrt {c d x + a e} c d} + \frac {4 \, {\left (c d x + 2 \, a e\right )} f g}{\sqrt {c d x + a e} c^{2} d^{2}} + \frac {2 \, {\left (c^{2} d^{2} x^{2} - 4 \, a c d e x - 8 \, a^{2} e^{2}\right )} g^{2}}{3 \, \sqrt {c d x + a e} c^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.43, size = 178, normalized size = 0.98 \[ -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (16\,a^2\,e^2\,g^2-24\,a\,c\,d\,e\,f\,g+6\,c^2\,d^2\,f^2\right )}{3\,c^4\,d^4\,e}-\frac {2\,g^2\,x^2\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e}+\frac {4\,g\,x\,\left (2\,a\,e\,g-3\,c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,c^3\,d^3\,e}\right )}{\frac {a}{c}+x^2+\frac {x\,\left (3\,c^4\,d^5+3\,a\,c^3\,d^3\,e^2\right )}{3\,c^4\,d^4\,e}} \]
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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