Optimal. Leaf size=191 \[ -\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-4 b e g+c d g+7 c e f)}{105 c^3 e^2 (d+e x)^{3/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-4 b e g+c d g+7 c e f)}{35 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2} \]
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Rubi [A] time = 0.30, antiderivative size = 191, 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 = {794, 656, 648} \begin {gather*} -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-4 b e g+c d g+7 c e f)}{35 c^2 e^2 \sqrt {d+e x}}-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-4 b e g+c d g+7 c e f)}{105 c^3 e^2 (d+e x)^{3/2}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int \sqrt {d+e x} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx &=-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2}-\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 \sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{7 c e^3}\\ &=-\frac {2 (7 c e f+c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{35 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2}+\frac {(2 (2 c d-b e) (7 c e f+c d g-4 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx}{35 c^2 e}\\ &=-\frac {4 (2 c d-b e) (7 c e f+c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{105 c^3 e^2 (d+e x)^{3/2}}-\frac {2 (7 c e f+c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{35 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 119, normalized size = 0.62 \begin {gather*} \frac {2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)} \left (8 b^2 e^2 g-2 b c e (15 d g+7 e f+6 e g x)+c^2 \left (22 d^2 g+d e (49 f+33 g x)+3 e^2 x (7 f+5 g x)\right )\right )}{105 c^3 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 139, normalized size = 0.73 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{3/2} \left (8 b^2 e^2 g-12 b c e g (d+e x)-18 b c d e g-14 b c e^2 f+4 c^2 d^2 g+21 c^2 e f (d+e x)+28 c^2 d e f+15 c^2 g (d+e x)^2+3 c^2 d g (d+e x)\right )}{105 c^3 e^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 233, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f + {\left (6 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} g\right )} x^{2} - 7 \, {\left (7 \, c^{3} d^{2} e - 9 \, b c^{2} d e^{2} + 2 \, b^{2} c e^{3}\right )} f - 2 \, {\left (11 \, c^{3} d^{3} - 26 \, b c^{2} d^{2} e + 19 \, b^{2} c d e^{2} - 4 \, b^{3} e^{3}\right )} g + {\left (7 \, {\left (4 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} f - {\left (11 \, c^{3} d^{2} e - 15 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\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}}{105 \, {\left (c^{3} e^{3} x + c^{3} d e^{2}\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 = 0.73 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (15 g \,x^{2} c^{2} e^{2}-12 b c \,e^{2} g x +33 c^{2} d e g x +21 c^{2} e^{2} f x +8 b^{2} e^{2} g -30 b c d e g -14 b c \,e^{2} f +22 c^{2} d^{2} g +49 c^{2} d e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{105 \sqrt {e x +d}\, c^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 236, normalized size = 1.24 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} e^{2} x^{2} - 7 \, c^{2} d^{2} + 9 \, b c d e - 2 \, b^{2} e^{2} + {\left (4 \, c^{2} d e + b c e^{2}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{15 \, {\left (c^{2} e^{2} x + c^{2} d e\right )}} + \frac {2 \, {\left (15 \, c^{3} e^{3} x^{3} - 22 \, c^{3} d^{3} + 52 \, b c^{2} d^{2} e - 38 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} + 3 \, {\left (6 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{2} - {\left (11 \, c^{3} d^{2} e - 15 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{105 \, {\left (c^{3} e^{3} x + c^{3} d e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.65, size = 219, normalized size = 1.15 \begin {gather*} \frac {\left (\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{7}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (b\,e\,g+6\,c\,d\,g+7\,c\,e\,f\right )}{35\,c\,e}+\frac {2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (8\,g\,b^2\,e^2-30\,g\,b\,c\,d\,e-14\,f\,b\,c\,e^2+22\,g\,c^2\,d^2+49\,f\,c^2\,d\,e\right )}{105\,c^3\,e^3}+\frac {x\,\sqrt {d+e\,x}\,\left (-8\,g\,b^2\,c\,e^3+30\,g\,b\,c^2\,d\,e^2+14\,f\,b\,c^2\,e^3-22\,g\,c^3\,d^2\,e+56\,f\,c^3\,d\,e^2\right )}{105\,c^3\,e^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x+\frac {d}{e}} \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 \sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \sqrt {d + e x} \left (f + g x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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