Optimal. Leaf size=217 \[ \frac {4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^3 e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^2 e^2 (2 c d-b e)}+\frac {2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]
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Rubi [A] time = 0.29, antiderivative size = 217, 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 = {788, 656, 648} \begin {gather*} \frac {2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^2 e^2 (2 c d-b e)}+\frac {4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^3 e^2 \sqrt {d+e x}}+\frac {2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
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 )^{3/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^{5/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(3 c e f+5 c d g-4 b e g) \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{5/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (3 c e f+5 c d g-4 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c^2 e^2 (2 c d-b e)}-\frac {(2 (3 c e f+5 c d g-4 b e g)) \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{3 c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{5/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {4 (3 c e f+5 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c^3 e^2 \sqrt {d+e x}}+\frac {2 (3 c e f+5 c d g-4 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c^2 e^2 (2 c d-b e)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 105, normalized size = 0.48 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (-8 b^2 e^2 g+2 b c e (11 d g+3 e f-2 e g x)+c^2 \left (-14 d^2 g+d e (7 g x-9 f)+e^2 x (3 f+g x)\right )\right )}{3 c^3 e^2 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.55, size = 155, normalized size = 0.71 \begin {gather*} \frac {2 \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} \left (-8 b^2 e^2 g-4 b c e g (d+e x)+26 b c d e g+6 b c e^2 f-20 c^2 d^2 g+3 c^2 e f (d+e x)-12 c^2 d e f+c^2 g (d+e x)^2+5 c^2 d g (d+e x)\right )}{3 c^3 e^2 \sqrt {d+e x} (b e+c (d+e x)-2 c d)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 165, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (c^{2} e^{2} g x^{2} - 3 \, {\left (3 \, c^{2} d e - 2 \, b c e^{2}\right )} f - 2 \, {\left (7 \, c^{2} d^{2} - 11 \, b c d e + 4 \, b^{2} e^{2}\right )} g + {\left (3 \, c^{2} e^{2} f + {\left (7 \, c^{2} d e - 4 \, b c e^{2}\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}}{3 \, {\left (c^{4} e^{4} x^{2} + b c^{3} e^{4} x - c^{4} d^{2} e^{2} + b c^{3} d e^{3}\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 = 139, normalized size = 0.64 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-g \,x^{2} c^{2} e^{2}+4 b c \,e^{2} g x -7 c^{2} d e g x -3 c^{2} e^{2} f x +8 b^{2} e^{2} g -22 b c d e g -6 b c \,e^{2} f +14 c^{2} d^{2} g +9 c^{2} d e f \right ) \left (e x +d \right )^{\frac {3}{2}}}{3 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}} c^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 112, normalized size = 0.52 \begin {gather*} -\frac {2 \, {\left (c e x - 3 \, c d + 2 \, b e\right )} f}{\sqrt {-c e x + c d - b e} c^{2} e} - \frac {2 \, {\left (c^{2} e^{2} x^{2} - 14 \, c^{2} d^{2} + 22 \, b c d e - 8 \, b^{2} e^{2} + {\left (7 \, c^{2} d e - 4 \, b c e^{2}\right )} x\right )} g}{3 \, \sqrt {-c e x + c d - b e} c^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.76, size = 167, normalized size = 0.77 \begin {gather*} \frac {\left (\frac {2\,g\,x^2\,\sqrt {d+e\,x}}{3\,c^2\,e^2}-\frac {\sqrt {d+e\,x}\,\left (16\,g\,b^2\,e^2-44\,g\,b\,c\,d\,e-12\,f\,b\,c\,e^2+28\,g\,c^2\,d^2+18\,f\,c^2\,d\,e\right )}{3\,c^4\,e^4}+\frac {2\,x\,\sqrt {d+e\,x}\,\left (7\,c\,d\,g-4\,b\,e\,g+3\,c\,e\,f\right )}{3\,c^3\,e^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x^2+\frac {b\,x}{c}+\frac {d\,\left (b\,e-c\,d\right )}{c\,e^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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