Optimal. Leaf size=118 \[ -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g-c d g+5 c e f)}{15 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt {d+e x}} \]
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Rubi [A] time = 0.17, antiderivative size = 118, 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 = {794, 648} \[ -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g-c d g+5 c e f)}{15 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rubi steps
\begin {align*} \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt {d+e x}}-\frac {\left (2 \left (\frac {3}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {1}{2} \left (c e^3 f-\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx}{5 c e^3}\\ &=-\frac {2 (5 c e f-c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 76, normalized size = 0.64 \[ \frac {2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)} (c (2 d g+5 e f+3 e g x)-2 b e g)}{15 c^2 e^2 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 140, normalized size = 1.19 \[ \frac {2 \, {\left (3 \, c^{2} e^{2} g x^{2} - 5 \, {\left (c^{2} d e - b c e^{2}\right )} f - 2 \, {\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} g + {\left (5 \, c^{2} e^{2} f - {\left (c^{2} d e - b c e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{15 \, {\left (c^{2} e^{3} x + c^{2} d e^{2}\right )}} \]
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 \[ \int \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{\sqrt {e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 79, normalized size = 0.67 \[ -\frac {2 \left (c e x +b e -c d \right ) \left (-3 c e g x +2 b e g -2 c d g -5 c e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{15 \sqrt {e x +d}\, c^{2} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 112, normalized size = 0.95 \[ \frac {2 \, {\left (c e x - c d + b e\right )} \sqrt {-c e x + c d - b e} f}{3 \, c e} + \frac {2 \, {\left (3 \, c^{2} e^{2} x^{2} - 2 \, c^{2} d^{2} + 4 \, b c d e - 2 \, b^{2} e^{2} - {\left (c^{2} d e - b c e^{2}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{15 \, c^{2} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.50, size = 100, normalized size = 0.85 \[ \frac {\left (\frac {2\,g\,x^2}{5}+\frac {2\,x\,\left (b\,e\,g-c\,d\,g+5\,c\,e\,f\right )}{15\,c\,e}+\frac {2\,\left (b\,e-c\,d\right )\,\left (2\,c\,d\,g-2\,b\,e\,g+5\,c\,e\,f\right )}{15\,c^2\,e^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{\sqrt {d+e\,x}} \]
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 \[ \int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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