Optimal. Leaf size=148 \[ \frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{c^2 e^2 \sqrt {d+e x} (2 c d-b e)}+\frac {2 (d+e x)^{3/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]
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Rubi [A] time = 0.16, antiderivative size = 148, 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 (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{c^2 e^2 \sqrt {d+e x} (2 c d-b e)}+\frac {2 (d+e x)^{3/2} (-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 788
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} (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)^{3/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(c e f+3 c d g-2 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}{c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{3/2}}{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 (c e f+3 c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^2 e^2 (2 c d-b e) \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 0.41 \begin {gather*} \frac {2 \sqrt {d+e x} (-2 b e g+2 c d g+c e (f-g x))}{c^2 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 = 3.68, size = 88, normalized size = 0.59 \begin {gather*} \frac {2 \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} (2 b e g+c g (d+e x)-3 c d g-c e f)}{c^2 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 = 102, normalized size = 0.69 \begin {gather*} \frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c e g x - c e f - 2 \, {\left (c d - b e\right )} g\right )} \sqrt {e x + d}}{c^{3} e^{4} x^{2} + b c^{2} e^{4} x - c^{3} d^{2} e^{2} + b c^{2} d e^{3}} \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 = 0.05, size = 78, normalized size = 0.53 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (c e g x +2 b e g -2 c d g -c e f \right ) \left (e x +d \right )^{\frac {3}{2}}}{\left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}} c^{2} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 62, normalized size = 0.42 \begin {gather*} \frac {2 \, f}{\sqrt {-c e x + c d - b e} c e} - \frac {2 \, {\left (c e x - 2 \, c d + 2 \, b e\right )} g}{\sqrt {-c e x + c d - b e} c^{2} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.78, size = 107, normalized size = 0.72 \begin {gather*} -\frac {\left (\frac {\sqrt {d+e\,x}\,\left (4\,c\,d\,g-4\,b\,e\,g+2\,c\,e\,f\right )}{c^3\,e^4}-\frac {2\,g\,x\,\sqrt {d+e\,x}}{c^2\,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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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