Optimal. Leaf size=217 \[ -\frac {4 \sqrt {d+e x} (-4 b e g+7 c d g+c e f)}{3 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{3/2} (-4 b e g+7 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.29, 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 (d+e x)^{3/2} (-4 b e g+7 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 {4 \sqrt {d+e x} (-4 b e g+7 c d g+c e f)}{3 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{7/2} (-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 656
Rule 788
Rubi steps
\begin {align*} \int \frac {(d+e x)^{7/2} (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)^{7/2}}{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+7 c d g-4 b e g) \int \frac {(d+e x)^{5/2}}{\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)^{7/2}}{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+7 c d g-4 b e g) (d+e x)^{3/2}}{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 (c e f+7 c d g-4 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^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{7/2}}{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 {4 (c e f+7 c d g-4 b e g) \sqrt {d+e x}}{3 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (c e f+7 c d g-4 b e g) (d+e x)^{3/2}}{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 = 0.09, size = 117, normalized size = 0.54 \begin {gather*} \frac {2 \sqrt {d+e x} \left (8 b^2 e^2 g-2 b c e (9 d g+e (f-6 g x))+c^2 \left (10 d^2 g+d e (f-15 g x)+3 e^2 x (g x-f)\right )\right )}{3 c^3 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 = 5.83, size = 139, normalized size = 0.64 \begin {gather*} -\frac {2 (d+e x)^{3/2} \left (8 b^2 e^2 g+12 b c e g (d+e x)-30 b c d e g-2 b c e^2 f+28 c^2 d^2 g-3 c^2 e f (d+e x)+4 c^2 d e f+3 c^2 g (d+e x)^2-21 c^2 d g (d+e x)\right )}{3 c^3 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 = 216, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (3 \, c^{2} e^{2} g x^{2} + {\left (c^{2} d e - 2 \, b c e^{2}\right )} f + 2 \, {\left (5 \, c^{2} d^{2} - 9 \, b c d e + 4 \, b^{2} e^{2}\right )} g - 3 \, {\left (c^{2} e^{2} f + {\left (5 \, 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^{5} e^{5} x^{3} + c^{5} d^{3} e^{2} - 2 \, b c^{4} d^{2} e^{3} + b^{2} c^{3} d e^{4} - {\left (c^{5} d e^{4} - 2 \, b c^{4} e^{5}\right )} x^{2} - {\left (c^{5} d^{2} e^{3} - b^{2} c^{3} 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 = 138, normalized size = 0.64 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (3 g \,x^{2} c^{2} e^{2}+12 b c \,e^{2} g x -15 c^{2} d e g x -3 c^{2} e^{2} f x +8 b^{2} e^{2} g -18 b c d e g -2 b c \,e^{2} f +10 c^{2} d^{2} g +c^{2} d 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^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 157, normalized size = 0.72 \begin {gather*} -\frac {2 \, {\left (3 \, c e x - c d + 2 \, b e\right )} f}{3 \, {\left (c^{3} e^{2} x - c^{3} d e + b c^{2} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (3 \, c^{2} e^{2} x^{2} + 10 \, c^{2} d^{2} - 18 \, b c d e + 8 \, b^{2} e^{2} - 3 \, {\left (5 \, c^{2} d e - 4 \, b c e^{2}\right )} x\right )} g}{3 \, {\left (c^{4} e^{3} x - c^{4} d e^{2} + b c^{3} e^{3}\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 = 2.89, size = 214, normalized size = 0.99 \begin {gather*} -\frac {\left (\frac {\sqrt {d+e\,x}\,\left (16\,g\,b^2\,e^2-36\,g\,b\,c\,d\,e-4\,f\,b\,c\,e^2+20\,g\,c^2\,d^2+2\,f\,c^2\,d\,e\right )}{3\,c^5\,e^5}+\frac {2\,g\,x^2\,\sqrt {d+e\,x}}{c^3\,e^3}-\frac {2\,x\,\sqrt {d+e\,x}\,\left (5\,c\,d\,g-4\,b\,e\,g+c\,e\,f\right )}{c^4\,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^3\,e^5-3\,c^5\,d^2\,e^3\right )}{3\,c^5\,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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