Optimal. Leaf size=155 \[ \frac {2 \sqrt {d+e x} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2 (2 c d-b e)^{3/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 155, 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, 660, 208} \[ \frac {2 \sqrt {d+e x} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2 (2 c d-b e)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 660
Rule 788
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x} (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) \sqrt {d+e x}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(e f-d g) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{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 (e f-d g)) \operatorname {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right )}{2 c d-b e}\\ &=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{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 (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2 (2 c d-b e)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 147, normalized size = 0.95 \[ \frac {2 \sqrt {d+e x} \left (c \sqrt {2 c d-b e} (d g-e f) \sqrt {c (d-e x)-b e} \tanh ^{-1}\left (\frac {\sqrt {-b e+c d-c e x}}{\sqrt {2 c d-b e}}\right )+(2 c d-b e) (-b e g+c d g+c e f)\right )}{c e^2 (b e-2 c d)^2 \sqrt {(d+e x) (c (d-e x)-b e)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 776, normalized size = 5.01 \[ \left [-\frac {{\left ({\left (c^{2} e^{3} f - c^{2} d e^{2} g\right )} x^{2} - {\left (c^{2} d^{2} e - b c d e^{2}\right )} f + {\left (c^{2} d^{3} - b c d^{2} e\right )} g + {\left (b c e^{3} f - b c d e^{2} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x + 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} f + {\left (2 \, c^{2} d^{2} - 3 \, b c d e + b^{2} e^{2}\right )} g\right )} \sqrt {e x + d}}{4 \, c^{4} d^{4} e^{2} - 8 \, b c^{3} d^{3} e^{3} + 5 \, b^{2} c^{2} d^{2} e^{4} - b^{3} c d e^{5} - {\left (4 \, c^{4} d^{2} e^{4} - 4 \, b c^{3} d e^{5} + b^{2} c^{2} e^{6}\right )} x^{2} - {\left (4 \, b c^{3} d^{2} e^{4} - 4 \, b^{2} c^{2} d e^{5} + b^{3} c e^{6}\right )} x}, \frac {2 \, {\left ({\left ({\left (c^{2} e^{3} f - c^{2} d e^{2} g\right )} x^{2} - {\left (c^{2} d^{2} e - b c d e^{2}\right )} f + {\left (c^{2} d^{3} - b c d^{2} e\right )} g + {\left (b c e^{3} f - b c d e^{2} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) + \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} f + {\left (2 \, c^{2} d^{2} - 3 \, b c d e + b^{2} e^{2}\right )} g\right )} \sqrt {e x + d}\right )}}{4 \, c^{4} d^{4} e^{2} - 8 \, b c^{3} d^{3} e^{3} + 5 \, b^{2} c^{2} d^{2} e^{4} - b^{3} c d e^{5} - {\left (4 \, c^{4} d^{2} e^{4} - 4 \, b c^{3} d e^{5} + b^{2} c^{2} e^{6}\right )} x^{2} - {\left (4 \, b c^{3} d^{2} e^{4} - 4 \, b^{2} c^{2} d e^{5} + b^{3} c e^{6}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 207, normalized size = 1.34 \[ -\frac {2 \left (\sqrt {-c e x -b e +c d}\, c d g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-\sqrt {-c e x -b e +c d}\, c e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+\sqrt {b e -2 c d}\, b e g -\sqrt {b e -2 c d}\, c d g -\sqrt {b e -2 c d}\, c e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{\left (b e -2 c d \right )^{\frac {3}{2}} \left (c e x +b e -c d \right ) \sqrt {e x +d}\, c \,e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x + d} {\left (g x + f\right )}}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (f+g\,x\right )\,\sqrt {d+e\,x}}{{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {d + e x} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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