Optimal. Leaf size=63 \[ \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)} \]
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Rubi [A] time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {860} \begin {gather*} \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 860
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 52, normalized size = 0.83 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.40, size = 168, normalized size = 2.67 \begin {gather*} -\frac {2 \left (\frac {(d g+e g x) (a e g+c d g x)}{g^2}\right )^{3/2} \sqrt {a e g+c d (f+g x)-c d f} \left (a^2 e^2 g^2+2 a c d e g (f+g x)-2 a c d e f g+c^2 d^2 f^2+c^2 d^2 (f+g x)^2-2 c^2 d^2 f (f+g x)\right )}{5 g (d+e x)^{3/2} (f+g x)^{5/2} (a e g-c d f) (a e g+c d g x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 232, normalized size = 3.68 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{5 \, {\left (c d^{2} f^{4} - a d e f^{3} g + {\left (c d e f g^{3} - a e^{2} g^{4}\right )} x^{4} + {\left (3 \, c d e f^{2} g^{2} - a d e g^{4} + {\left (c d^{2} - 3 \, a e^{2}\right )} f g^{3}\right )} x^{3} + 3 \, {\left (c d e f^{3} g - a d e f g^{3} + {\left (c d^{2} - a e^{2}\right )} f^{2} g^{2}\right )} x^{2} + {\left (c d e f^{4} - 3 \, a d e f^{2} g^{2} + {\left (3 \, c d^{2} - a e^{2}\right )} f^{3} g\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 = 63, normalized size = 1.00 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{5 \left (g x +f \right )^{\frac {5}{2}} \left (a e g -c d f \right ) \left (e x +d \right )^{\frac {3}{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 (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {7}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.07, size = 232, normalized size = 3.68 \begin {gather*} -\frac {\left (\frac {2\,a^2\,e^2}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {2\,c^2\,d^2\,x^2}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {4\,a\,c\,d\,e\,x}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}-\frac {\sqrt {f+g\,x}\,\left (5\,c\,d\,f^3-5\,a\,e\,f^2\,g\right )\,\sqrt {d+e\,x}}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {x\,\sqrt {f+g\,x}\,\left (10\,a\,e\,f\,g^2-10\,c\,d\,f^2\,g\right )\,\sqrt {d+e\,x}}{5\,a\,e\,g^3-5\,c\,d\,f\,g^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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