Optimal. Leaf size=152 \[ \frac {2 \sqrt {d+e x} (2 b e g-5 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 152, 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 = {788, 648} \begin {gather*} \frac {2 \sqrt {d+e x} (2 b e g-5 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 788
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^{5/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(c e f-5 c d g+2 b e g) \int \frac {(d+e x)^{3/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{5/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 (c e f-5 c d g+2 b e g) \sqrt {d+e x}}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 76, normalized size = 0.50 \begin {gather*} -\frac {2 \sqrt {d+e x} (2 b e g-2 c d g+c e (f+3 g x))}{3 c^2 e^2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 6.22, size = 73, normalized size = 0.48 \begin {gather*} \frac {2 (d+e x)^{3/2} (2 b e g+3 c g (d+e x)-5 c d g+c e f)}{3 c^2 e^2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 154, normalized size = 1.01 \begin {gather*} \frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (3 \, c e g x + c e f - 2 \, {\left (c d - b e\right )} g\right )} \sqrt {e x + d}}{3 \, {\left (c^{4} e^{5} x^{3} + c^{4} d^{3} e^{2} - 2 \, b c^{3} d^{2} e^{3} + b^{2} c^{2} d e^{4} - {\left (c^{4} d e^{4} - 2 \, b c^{3} e^{5}\right )} x^{2} - {\left (c^{4} d^{2} e^{3} - b^{2} c^{2} e^{5}\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 = 0.04, size = 78, normalized size = 0.51 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (3 c e g x +2 b e g -2 c d g +c e f \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\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 [A] time = 0.97, size = 103, normalized size = 0.68 \begin {gather*} -\frac {2 \, {\left (3 \, c e x - 2 \, c d + 2 \, b e\right )} g}{3 \, {\left (c^{3} e^{3} x - c^{3} d e^{2} + b c^{2} e^{3}\right )} \sqrt {-c e x + c d - b e}} - \frac {2 \, f}{3 \, {\left (c^{2} e^{2} x - c^{2} d e + b c e^{2}\right )} \sqrt {-c e x + c d - b e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.85, size = 154, normalized size = 1.01 \begin {gather*} \frac {\left (\frac {\sqrt {d+e\,x}\,\left (4\,b\,e\,g-4\,c\,d\,g+2\,c\,e\,f\right )}{3\,c^4\,e^5}+\frac {2\,g\,x\,\sqrt {d+e\,x}}{c^3\,e^4}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x^3+\frac {x\,\left (3\,b^2\,c^2\,e^5-3\,c^4\,d^2\,e^3\right )}{3\,c^4\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \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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