Optimal. Leaf size=125 \[ \frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d e (d+e x)^{5/2}}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{63 c^2 d^2 e (d+e x)^{7/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {794, 648} \begin {gather*} \frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d e (d+e x)^{5/2}}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{63 c^2 d^2 e (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=\frac {2 g \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d e (d+e x)^{5/2}}+\frac {1}{9} \left (9 f-\frac {7 d g}{e}-\frac {2 a e g}{c d}\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx\\ &=\frac {2 \left (9 f-\frac {7 d g}{e}-\frac {2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{63 c d (d+e x)^{7/2}}+\frac {2 g \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d e (d+e x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 64, normalized size = 0.51 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} (c d (9 f+7 g x)-2 a e g)}{63 c^2 d^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.73, size = 67, normalized size = 0.54 \begin {gather*} \frac {2 (a e+c d x) ((d+e x) (a e+c d x))^{5/2} (7 g (a e+c d x)-9 a e g+9 c d f)}{63 c^2 d^2 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 173, normalized size = 1.38 \begin {gather*} \frac {2 \, {\left (7 \, c^{4} d^{4} g x^{4} + 9 \, a^{3} c d e^{3} f - 2 \, a^{4} e^{4} g + {\left (9 \, c^{4} d^{4} f + 19 \, a c^{3} d^{3} e g\right )} x^{3} + 3 \, {\left (9 \, a c^{3} d^{3} e f + 5 \, a^{2} c^{2} d^{2} e^{2} g\right )} x^{2} + {\left (27 \, a^{2} c^{2} d^{2} e^{2} f + a^{3} c d e^{3} g\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}}{63 \, {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 67, normalized size = 0.54 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-7 c d g x +2 a e g -9 c d f \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {5}{2}} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 141, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d x + a e} f}{7 \, c d} + \frac {2 \, {\left (7 \, c^{4} d^{4} x^{4} + 19 \, a c^{3} d^{3} e x^{3} + 15 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{3} c d e^{3} x - 2 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} g}{63 \, c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.37, size = 134, normalized size = 1.07 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,c^2\,d^2\,g\,x^4}{9}+\frac {2\,a\,e\,x^2\,\left (5\,a\,e\,g+9\,c\,d\,f\right )}{21}+\frac {2\,c\,d\,x^3\,\left (19\,a\,e\,g+9\,c\,d\,f\right )}{63}-\frac {2\,a^3\,e^3\,\left (2\,a\,e\,g-9\,c\,d\,f\right )}{63\,c^2\,d^2}+\frac {2\,a^2\,e^2\,x\,\left (a\,e\,g+27\,c\,d\,f\right )}{63\,c\,d}\right )}{\sqrt {d+e\,x}} \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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