Optimal. Leaf size=291 \[ -\frac {16 \sqrt {d+e x} (2 c d-b e) (-2 b e g+3 c d g+c e f)}{3 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {8 (d+e x)^{3/2} (-2 b e g+3 c d g+c e f)}{3 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{5/2} (-2 b e g+3 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.41, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {788, 656, 648} \begin {gather*} \frac {2 (d+e x)^{5/2} (-2 b e g+3 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {8 (d+e x)^{3/2} (-2 b e g+3 c d g+c e f)}{3 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {16 \sqrt {d+e x} (2 c d-b e) (-2 b e g+3 c d g+c e f)}{3 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 788
Rubi steps
\begin {align*} \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(c e f+3 c d g-2 b e g) \int \frac {(d+e x)^{7/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 (c e f+3 c d g-2 b e g) (d+e x)^{5/2}}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(4 (c e f+3 c d g-2 b e g)) \int \frac {(d+e x)^{5/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {8 (c e f+3 c d g-2 b e g) (d+e x)^{3/2}}{3 c^3 e^2 \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) (d+e x)^{5/2}}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(8 (2 c d-b e) (c e f+3 c d g-2 b e g)) \int \frac {(d+e x)^{3/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {16 (2 c d-b e) (c e f+3 c d g-2 b e g) \sqrt {d+e x}}{3 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {8 (c e f+3 c d g-2 b e g) (d+e x)^{3/2}}{3 c^3 e^2 \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) (d+e x)^{5/2}}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 180, normalized size = 0.62 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-16 b^3 e^3 g+8 b^2 c e^2 (8 d g+e (f-3 g x))-2 b c^2 e \left (41 d^2 g+2 d e (5 f-18 g x)+3 e^2 x (g x-2 f)\right )+c^3 \left (34 d^3 g+d^2 e (11 f-51 g x)+6 d e^2 x (2 g x-3 f)+e^3 x^2 (3 f+g x)\right )\right )}{3 c^4 e^2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.96, size = 247, normalized size = 0.85 \begin {gather*} -\frac {2 (d+e x)^{3/2} \left (-16 b^3 e^3 g-24 b^2 c e^2 g (d+e x)+88 b^2 c d e^2 g+8 b^2 c e^3 f-160 b c^2 d^2 e g+12 b c^2 e^2 f (d+e x)-32 b c^2 d e^2 f-6 b c^2 e g (d+e x)^2+84 b c^2 d e g (d+e x)+96 c^3 d^3 g+32 c^3 d^2 e f-72 c^3 d^2 g (d+e x)+3 c^3 e f (d+e x)^2-24 c^3 d e f (d+e x)+c^3 g (d+e x)^3+9 c^3 d g (d+e x)^2\right )}{3 c^4 e^2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 308, normalized size = 1.06 \begin {gather*} -\frac {2 \, {\left (c^{3} e^{3} g x^{3} + 3 \, {\left (c^{3} e^{3} f + 2 \, {\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} g\right )} x^{2} + {\left (11 \, c^{3} d^{2} e - 20 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f + 2 \, {\left (17 \, c^{3} d^{3} - 41 \, b c^{2} d^{2} e + 32 \, b^{2} c d e^{2} - 8 \, b^{3} e^{3}\right )} g - 3 \, {\left (2 \, {\left (3 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f + {\left (17 \, c^{3} d^{2} e - 24 \, b c^{2} d e^{2} + 8 \, 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}}{3 \, {\left (c^{6} e^{5} x^{3} + c^{6} d^{3} e^{2} - 2 \, b c^{5} d^{2} e^{3} + b^{2} c^{4} d e^{4} - {\left (c^{6} d e^{4} - 2 \, b c^{5} e^{5}\right )} x^{2} - {\left (c^{6} d^{2} e^{3} - b^{2} c^{4} e^{5}\right )} x\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 = 235, normalized size = 0.81 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-g \,e^{3} x^{3} c^{3}+6 b \,c^{2} e^{3} g \,x^{2}-12 c^{3} d \,e^{2} g \,x^{2}-3 c^{3} e^{3} f \,x^{2}+24 b^{2} c \,e^{3} g x -72 b \,c^{2} d \,e^{2} g x -12 b \,c^{2} e^{3} f x +51 c^{3} d^{2} e g x +18 c^{3} d \,e^{2} f x +16 b^{3} e^{3} g -64 b^{2} c d \,e^{2} g -8 b^{2} c \,e^{3} f +82 b \,c^{2} d^{2} e g +20 b \,c^{2} d \,e^{2} f -34 c^{3} d^{3} g -11 f \,d^{2} c^{3} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}} c^{4} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 246, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} e^{2} x^{2} + 11 \, c^{2} d^{2} - 20 \, b c d e + 8 \, b^{2} e^{2} - 6 \, {\left (3 \, c^{2} d e - 2 \, b c e^{2}\right )} x\right )} f}{3 \, {\left (c^{4} e^{2} x - c^{4} d e + b c^{3} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (c^{3} e^{3} x^{3} + 34 \, c^{3} d^{3} - 82 \, b c^{2} d^{2} e + 64 \, b^{2} c d e^{2} - 16 \, b^{3} e^{3} + 6 \, {\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} - 3 \, {\left (17 \, c^{3} d^{2} e - 24 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} g}{3 \, {\left (c^{5} e^{3} x - c^{5} d e^{2} + b c^{4} e^{3}\right )} \sqrt {-c e x + c d - b e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.21, size = 314, normalized size = 1.08 \begin {gather*} -\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {\sqrt {d+e\,x}\,\left (-32\,g\,b^3\,e^3+128\,g\,b^2\,c\,d\,e^2+16\,f\,b^2\,c\,e^3-164\,g\,b\,c^2\,d^2\,e-40\,f\,b\,c^2\,d\,e^2+68\,g\,c^3\,d^3+22\,f\,c^3\,d^2\,e\right )}{3\,c^6\,e^5}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (4\,c\,d\,g-2\,b\,e\,g+c\,e\,f\right )}{c^4\,e^3}+\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{3\,c^3\,e^2}-\frac {x\,\sqrt {d+e\,x}\,\left (48\,g\,b^2\,c\,e^3-144\,g\,b\,c^2\,d\,e^2-24\,f\,b\,c^2\,e^3+102\,g\,c^3\,d^2\,e+36\,f\,c^3\,d\,e^2\right )}{3\,c^6\,e^5}\right )}{x^3+\frac {x\,\left (3\,b^2\,c^4\,e^5-3\,c^6\,d^2\,e^3\right )}{3\,c^6\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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