Optimal. Leaf size=118 \[ -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-2 b e g-5 c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{7/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2 (d+e x)^{5/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {794, 648} \begin {gather*} -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-2 b e g-5 c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{7/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2 (d+e x)^{5/2}}-\frac {\left (2 \left (\frac {7}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac {5}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{9 c e^3}\\ &=-\frac {2 (9 c e f-5 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 c^2 e^2 (d+e x)^{7/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2 (d+e x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 78, normalized size = 0.66 \begin {gather*} \frac {2 (b e-c d+c e x)^3 \sqrt {(d+e x) (c (d-e x)-b e)} (c (2 d g+9 e f+7 e g x)-2 b e g)}{63 c^2 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.66, size = 74, normalized size = 0.63 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{7/2} (-2 b e g+7 c g (d+e x)-5 c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 345, normalized size = 2.92 \begin {gather*} \frac {2 \, {\left (7 \, c^{4} e^{4} g x^{4} + {\left (9 \, c^{4} e^{4} f - 19 \, {\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} g\right )} x^{3} - 3 \, {\left (9 \, {\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} f - 5 \, {\left (c^{4} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} g\right )} x^{2} - 9 \, {\left (c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f - 2 \, {\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} g + {\left (27 \, {\left (c^{4} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} f - {\left (c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{63 \, {\left (c^{2} e^{3} x + c^{2} d e^{2}\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 = 0.05, size = 79, normalized size = 0.67 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-7 c e g x +2 b e g -2 c d g -9 c e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {5}{2}} c^{2} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.74, size = 314, normalized size = 2.66 \begin {gather*} \frac {2 \, {\left (c^{3} e^{3} x^{3} - c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + b^{3} e^{3} - 3 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} + 3 \, {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x\right )} \sqrt {-c e x + c d - b e} f}{7 \, c e} + \frac {2 \, {\left (7 \, c^{4} e^{4} x^{4} - 2 \, c^{4} d^{4} + 8 \, b c^{3} d^{3} e - 12 \, b^{2} c^{2} d^{2} e^{2} + 8 \, b^{3} c d e^{3} - 2 \, b^{4} e^{4} - 19 \, {\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + 15 \, {\left (c^{4} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{2} - {\left (c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{63 \, c^{2} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.90, size = 170, normalized size = 1.44 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\left (5\,b\,e\,g-5\,c\,d\,g+9\,c\,e\,f\right )}{21}+\frac {2\,c\,e\,x^3\,\left (19\,b\,e\,g-19\,c\,d\,g+9\,c\,e\,f\right )}{63}+\frac {2\,c^2\,e^2\,g\,x^4}{9}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\left (2\,c\,d\,g-2\,b\,e\,g+9\,c\,e\,f\right )}{63\,c^2\,e^2}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e\,g-c\,d\,g+27\,c\,e\,f\right )}{63\,c\,e}\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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