Optimal. Leaf size=193 \[ -\frac {4 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+3 c d g+5 c e f)}{15 c^3 e^2 \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+3 c d g+5 c e f)}{15 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e^2} \]
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Rubi [A] time = 0.32, antiderivative size = 193, 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 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+3 c d g+5 c e f)}{15 c^2 e^2}-\frac {4 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+3 c d g+5 c e f)}{15 c^3 e^2 \sqrt {d+e x}}-\frac {2 g (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {2 g (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e^2}-\frac {\left (2 \left (\frac {1}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {3}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{5 c e^3}\\ &=-\frac {2 (5 c e f+3 c d g-4 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e^2}+\frac {(2 (2 c d-b e) (5 c e f+3 c d g-4 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}{15 c^2 e}\\ &=-\frac {4 (2 c d-b e) (5 c e f+3 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^3 e^2 \sqrt {d+e x}}-\frac {2 (5 c e f+3 c d g-4 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 118, normalized size = 0.61 \begin {gather*} \frac {2 \sqrt {d+e x} (b e-c d+c e x) \left (8 b^2 e^2 g-2 b c e (13 d g+5 e f+2 e g x)+c^2 \left (18 d^2 g+d e (25 f+9 g x)+e^2 x (5 f+3 g x)\right )\right )}{15 c^3 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 = 0.23, size = 139, normalized size = 0.72 \begin {gather*} -\frac {2 \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} \left (8 b^2 e^2 g-4 b c e g (d+e x)-22 b c d e g-10 b c e^2 f+12 c^2 d^2 g+5 c^2 e f (d+e x)+20 c^2 d e f+3 c^2 g (d+e x)^2+3 c^2 d g (d+e x)\right )}{15 c^3 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 143, normalized size = 0.74 \begin {gather*} -\frac {2 \, {\left (3 \, c^{2} e^{2} g x^{2} + 5 \, {\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} f + 2 \, {\left (9 \, c^{2} d^{2} - 13 \, b c d e + 4 \, b^{2} e^{2}\right )} g + {\left (5 \, c^{2} e^{2} f + {\left (9 \, 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}}{15 \, {\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 = 0.05, size = 139, normalized size = 0.72 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (3 g \,x^{2} c^{2} e^{2}-4 b c \,e^{2} g x +9 c^{2} d e g x +5 c^{2} e^{2} f x +8 b^{2} e^{2} g -26 b c d e g -10 b c \,e^{2} f +18 c^{2} d^{2} g +25 c^{2} d e f \right ) \sqrt {e x +d}}{15 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 201, normalized size = 1.04 \begin {gather*} \frac {2 \, {\left (c^{2} e^{2} x^{2} - 5 \, c^{2} d^{2} + 7 \, b c d e - 2 \, b^{2} e^{2} + {\left (4 \, c^{2} d e - b c e^{2}\right )} x\right )} f}{3 \, \sqrt {-c e x + c d - b e} c^{2} e} + \frac {2 \, {\left (3 \, c^{3} e^{3} x^{3} - 18 \, c^{3} d^{3} + 44 \, b c^{2} d^{2} e - 34 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} + {\left (6 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} + {\left (9 \, c^{3} d^{2} e - 13 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\right )} x\right )} g}{15 \, \sqrt {-c e x + c d - b e} c^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.55, size = 149, normalized size = 0.77 \begin {gather*} -\frac {\left (\frac {\sqrt {d+e\,x}\,\left (16\,g\,b^2\,e^2-52\,g\,b\,c\,d\,e-20\,f\,b\,c\,e^2+36\,g\,c^2\,d^2+50\,f\,c^2\,d\,e\right )}{15\,c^3\,e^3}+\frac {2\,g\,x^2\,\sqrt {d+e\,x}}{5\,c\,e}+\frac {2\,x\,\sqrt {d+e\,x}\,\left (9\,c\,d\,g-4\,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}}{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 \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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