Optimal. Leaf size=239 \[ -\frac {16 g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^4 d^4 e \sqrt {d+e x}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^3 d^3 e}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.28, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {866, 794, 648} \[ -\frac {16 g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^4 d^4 e \sqrt {d+e x}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^3 d^3 e}-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rule 866
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(2 g) \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}{c d}\\ &=-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (8 g^2\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}{c^2 d^2}\\ &=-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^3 d^3 e}-\frac {\left (8 g^2 \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^3 d^3 e}\\ &=-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {16 g^2 \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^4 d^4 e \sqrt {d+e x}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^3 d^3 e}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 131, normalized size = 0.55 \[ \frac {2 (d+e x)^{3/2} \left (-16 a^3 e^3 g^3+24 a^2 c d e^2 g^2 (f-g x)-6 a c^2 d^2 e g \left (f^2-6 f g x+g^2 x^2\right )+c^3 d^3 \left (-f^3-9 f^2 g x+9 f g^2 x^2+g^3 x^3\right )\right )}{3 c^4 d^4 ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 251, normalized size = 1.05 \[ \frac {2 \, {\left (c^{3} d^{3} g^{3} x^{3} - c^{3} d^{3} f^{3} - 6 \, a c^{2} d^{2} e f^{2} g + 24 \, a^{2} c d e^{2} f g^{2} - 16 \, a^{3} e^{3} g^{3} + 3 \, {\left (3 \, c^{3} d^{3} f g^{2} - 2 \, a c^{2} d^{2} e g^{3}\right )} x^{2} - 3 \, {\left (3 \, c^{3} d^{3} f^{2} g - 12 \, a c^{2} d^{2} e f g^{2} + 8 \, a^{2} c d e^{2} g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3 \, {\left (c^{6} d^{6} e x^{3} + a^{2} c^{4} d^{5} e^{2} + {\left (c^{6} d^{7} + 2 \, a c^{5} d^{5} e^{2}\right )} x^{2} + {\left (2 \, a c^{5} d^{6} e + a^{2} c^{4} d^{4} e^{3}\right )} x\right )}} \]
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 = 0.78 \[ -\frac {2 \left (c d x +a e \right ) \left (-g^{3} x^{3} c^{3} d^{3}+6 a \,c^{2} d^{2} e \,g^{3} x^{2}-9 c^{3} d^{3} f \,g^{2} x^{2}+24 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x +9 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-24 a^{2} c d \,e^{2} f \,g^{2}+6 a \,c^{2} d^{2} e \,f^{2} g +f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}} c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 219, normalized size = 0.92 \[ -\frac {2 \, {\left (3 \, c d x + 2 \, a e\right )} f^{2} g}{{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )} \sqrt {c d x + a e}} + \frac {2 \, {\left (3 \, c^{2} d^{2} x^{2} + 12 \, a c d e x + 8 \, a^{2} e^{2}\right )} f g^{2}}{{\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )} \sqrt {c d x + a e}} + \frac {2 \, {\left (c^{3} d^{3} x^{3} - 6 \, a c^{2} d^{2} e x^{2} - 24 \, a^{2} c d e^{2} x - 16 \, a^{3} e^{3}\right )} g^{3}}{3 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )} \sqrt {c d x + a e}} - \frac {2 \, f^{3}}{3 \, {\left (c^{2} d^{2} x + a c d e\right )} \sqrt {c d x + a e}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.77, size = 278, normalized size = 1.16 \[ -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (\frac {32\,a^3\,e^3\,g^3}{3}-16\,a^2\,c\,d\,e^2\,f\,g^2+4\,a\,c^2\,d^2\,e\,f^2\,g+\frac {2\,c^3\,d^3\,f^3}{3}\right )}{c^6\,d^6\,e}-\frac {2\,g^3\,x^3\,\sqrt {d+e\,x}}{3\,c^3\,d^3\,e}+\frac {g^2\,x^2\,\left (4\,a\,e\,g-6\,c\,d\,f\right )\,\sqrt {d+e\,x}}{c^4\,d^4\,e}+\frac {2\,g\,x\,\sqrt {d+e\,x}\,\left (8\,a^2\,e^2\,g^2-12\,a\,c\,d\,e\,f\,g+3\,c^2\,d^2\,f^2\right )}{c^5\,d^5\,e}\right )}{x^3+\frac {a^2\,e}{c^2\,d}+\frac {a\,x\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}+\frac {x^2\,\left (c^6\,d^7+2\,a\,c^5\,d^5\,e^2\right )}{c^6\,d^6\,e}} \]
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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