Optimal. Leaf size=128 \[ \frac {4 g \sqrt {d+e x} \sqrt {f+g x}}{3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac {2 (d+e x)^{3/2} \sqrt {f+g x}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]
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Rubi [A] time = 0.14, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {868, 860} \begin {gather*} \frac {4 g \sqrt {d+e x} \sqrt {f+g x}}{3 \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}-\frac {2 (d+e x)^{3/2} \sqrt {f+g x}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 860
Rule 868
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\sqrt {f+g x} \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} \sqrt {f+g x}}{3 (c d f-a e g) \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}}{\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 (c d f-a e g)}\\ &=-\frac {2 (d+e x)^{3/2} \sqrt {f+g x}}{3 (c d f-a e g) \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} \sqrt {f+g x}}{3 (c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 68, normalized size = 0.53 \begin {gather*} \frac {2 (d+e x)^{3/2} \sqrt {f+g x} (3 a e g-c d (f-2 g x))}{3 ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.03, size = 119, normalized size = 0.93 \begin {gather*} \frac {2 (d+e x)^{5/2} \sqrt {f+g x} (a e g+c d g x)^{5/2} (3 a e g+2 c d (f+g x)-3 c d f)}{3 g (c d f-a e g)^2 \left (\frac {(d g+e g x) (a e g+c d g x)}{g^2}\right )^{5/2} (a e g+c d (f+g x)-c d f)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 318, normalized size = 2.48 \begin {gather*} \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x - c d f + 3 \, a e g\right )} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (a^{2} c^{2} d^{3} e^{2} f^{2} - 2 \, a^{3} c d^{2} e^{3} f g + a^{4} d e^{4} g^{2} + {\left (c^{4} d^{4} e f^{2} - 2 \, a c^{3} d^{3} e^{2} f g + a^{2} c^{2} d^{2} e^{3} g^{2}\right )} x^{3} + {\left ({\left (c^{4} d^{5} + 2 \, a c^{3} d^{3} e^{2}\right )} f^{2} - 2 \, {\left (a c^{3} d^{4} e + 2 \, a^{2} c^{2} d^{2} e^{3}\right )} f g + {\left (a^{2} c^{2} d^{3} e^{2} + 2 \, a^{3} c d e^{4}\right )} g^{2}\right )} x^{2} + {\left ({\left (2 \, a c^{3} d^{4} e + a^{2} c^{2} d^{2} e^{3}\right )} f^{2} - 2 \, {\left (2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} f g + {\left (2 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} g^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 99, normalized size = 0.77 \begin {gather*} \frac {2 \sqrt {g x +f}\, \left (c d x +a e \right ) \left (2 c d g x +3 a e g -c d f \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} \sqrt {g x + f}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.06, size = 246, normalized size = 1.92 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,g^2\,x^2\,\sqrt {d+e\,x}}{3\,c\,d\,e\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {\left (2\,c\,d\,f^2-6\,a\,e\,f\,g\right )\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {x\,\left (6\,a\,e\,g^2+2\,c\,d\,f\,g\right )\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )}{x^3\,\sqrt {f+g\,x}+\frac {a^2\,e\,\sqrt {f+g\,x}}{c^2\,d}+\frac {x^2\,\sqrt {f+g\,x}\,\left (c\,d^2+2\,a\,e^2\right )}{c\,d\,e}+\frac {a\,x\,\sqrt {f+g\,x}\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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