Optimal. Leaf size=193 \[ -\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{693 c^3 e^2 (d+e x)^{7/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{99 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}} \]
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Rubi [A] time = 0.34, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {794, 656, 648} \begin {gather*} -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{99 c^2 e^2 (d+e x)^{5/2}}-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{693 c^3 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}}{11 c e^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
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)^{3/2}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}-\frac {\left (2 \left (\frac {7}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac {3}{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)^{3/2}} \, dx}{11 c e^3}\\ &=-\frac {2 (11 c e f-3 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{99 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}+\frac {(2 (2 c d-b e) (11 c e f-3 c d g-4 b e g)) \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}{99 c^2 e}\\ &=-\frac {4 (2 c d-b e) (11 c e f-3 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{693 c^3 e^2 (d+e x)^{7/2}}-\frac {2 (11 c e f-3 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{99 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 121, normalized size = 0.63 \begin {gather*} \frac {2 (b e-c d+c e x)^3 \sqrt {(d+e x) (c (d-e x)-b e)} \left (8 b^2 e^2 g-2 b c e (19 d g+11 e f+14 e g x)+c^2 \left (30 d^2 g+d e (121 f+105 g x)+7 e^2 x (11 f+9 g x)\right )\right )}{693 c^3 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.16, size = 139, normalized size = 0.72 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{7/2} \left (8 b^2 e^2 g-28 b c e g (d+e x)-10 b c d e g-22 b c e^2 f-12 c^2 d^2 g+77 c^2 e f (d+e x)+44 c^2 d e f+63 c^2 g (d+e x)^2-21 c^2 d g (d+e x)\right )}{693 c^3 e^2 (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 500, normalized size = 2.59 \begin {gather*} \frac {2 \, {\left (63 \, c^{5} e^{5} g x^{5} + 7 \, {\left (11 \, c^{5} e^{5} f - {\left (12 \, c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} g\right )} x^{4} - {\left (11 \, {\left (10 \, c^{5} d e^{4} - 19 \, b c^{4} e^{5}\right )} f + {\left (96 \, c^{5} d^{2} e^{3} + 17 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} - 3 \, {\left (11 \, {\left (4 \, c^{5} d^{2} e^{3} + b c^{4} d e^{4} - 5 \, b^{2} c^{3} e^{5}\right )} f - {\left (54 \, c^{5} d^{3} e^{2} - 107 \, b c^{4} d^{2} e^{3} + 52 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} g\right )} x^{2} - 11 \, {\left (11 \, c^{5} d^{4} e - 35 \, b c^{4} d^{3} e^{2} + 39 \, b^{2} c^{3} d^{2} e^{3} - 17 \, b^{3} c^{2} d e^{4} + 2 \, b^{4} c e^{5}\right )} f - 2 \, {\left (15 \, c^{5} d^{5} - 64 \, b c^{4} d^{4} e + 106 \, b^{2} c^{3} d^{3} e^{2} - 84 \, b^{3} c^{2} d^{2} e^{3} + 31 \, b^{4} c d e^{4} - 4 \, b^{5} e^{5}\right )} g + {\left (11 \, {\left (26 \, c^{5} d^{3} e^{2} - 51 \, b c^{4} d^{2} e^{3} + 24 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} f - {\left (15 \, c^{5} d^{4} e - 49 \, b c^{4} d^{3} e^{2} + 57 \, b^{2} c^{3} d^{2} e^{3} - 27 \, b^{3} c^{2} d e^{4} + 4 \, b^{4} c e^{5}\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}}{693 \, {\left (c^{3} e^{3} x + c^{3} d e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 139, normalized size = 0.72 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (63 g \,x^{2} c^{2} e^{2}-28 b c \,e^{2} g x +105 c^{2} d e g x +77 c^{2} e^{2} f x +8 b^{2} e^{2} g -38 b c d e g -22 b c \,e^{2} f +30 c^{2} d^{2} g +121 c^{2} d e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{693 \left (e x +d \right )^{\frac {5}{2}} c^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.89, size = 465, normalized size = 2.41 \begin {gather*} \frac {2 \, {\left (7 \, c^{4} e^{4} x^{4} - 11 \, c^{4} d^{4} + 35 \, b c^{3} d^{3} e - 39 \, b^{2} c^{2} d^{2} e^{2} + 17 \, b^{3} c d e^{3} - 2 \, b^{4} e^{4} - {\left (10 \, c^{4} d e^{3} - 19 \, b c^{3} e^{4}\right )} x^{3} - 3 \, {\left (4 \, c^{4} d^{2} e^{2} + b c^{3} d e^{3} - 5 \, b^{2} c^{2} e^{4}\right )} x^{2} + {\left (26 \, c^{4} d^{3} e - 51 \, b c^{3} d^{2} e^{2} + 24 \, b^{2} c^{2} d e^{3} + b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} f}{63 \, c^{2} e} + \frac {2 \, {\left (63 \, c^{5} e^{5} x^{5} - 30 \, c^{5} d^{5} + 128 \, b c^{4} d^{4} e - 212 \, b^{2} c^{3} d^{3} e^{2} + 168 \, b^{3} c^{2} d^{2} e^{3} - 62 \, b^{4} c d e^{4} + 8 \, b^{5} e^{5} - 7 \, {\left (12 \, c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} x^{4} - {\left (96 \, c^{5} d^{2} e^{3} + 17 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \, {\left (54 \, c^{5} d^{3} e^{2} - 107 \, b c^{4} d^{2} e^{3} + 52 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} x^{2} - {\left (15 \, c^{5} d^{4} e - 49 \, b c^{4} d^{3} e^{2} + 57 \, b^{2} c^{3} d^{2} e^{3} - 27 \, b^{3} c^{2} d e^{4} + 4 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{693 \, c^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.23, size = 320, normalized size = 1.66 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,c^2\,e^3\,g\,x^5}{11}+\frac {2\,c\,e^2\,x^4\,\left (23\,b\,e\,g-12\,c\,d\,g+11\,c\,e\,f\right )}{99}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\left (g\,b^2\,e^2+53\,g\,b\,c\,d\,e+55\,f\,b\,c\,e^2-54\,g\,c^2\,d^2+44\,f\,c^2\,d\,e\right )}{231\,c}-\frac {x^3\,\left (-226\,g\,b^2\,c^3\,e^5+34\,g\,b\,c^4\,d\,e^4-418\,f\,b\,c^4\,e^5+192\,g\,c^5\,d^2\,e^3+220\,f\,c^5\,d\,e^4\right )}{693\,c^3\,e^2}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\left (8\,g\,b^2\,e^2-38\,g\,b\,c\,d\,e-22\,f\,b\,c\,e^2+30\,g\,c^2\,d^2+121\,f\,c^2\,d\,e\right )}{693\,c^3\,e^2}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\left (-4\,g\,b^2\,e^2+19\,g\,b\,c\,d\,e+11\,f\,b\,c\,e^2-15\,g\,c^2\,d^2+286\,f\,c^2\,d\,e\right )}{693\,c^2\,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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