Optimal. Leaf size=165 \[ \frac {2 (b+2 c x) (-b e g-2 c d g+4 c e f)}{3 c e (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (e x (2 c d-b e)+d (2 c d-b e)) (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {777, 613} \begin {gather*} \frac {2 (b+2 c x) (-b e g-2 c d g+4 c e f)}{3 c e (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (e x (2 c d-b e)+d (2 c d-b e)) (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e)^2 \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 613
Rule 777
Rubi steps
\begin {align*} \int \frac {(d+e x) (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)}{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 {(4 c e f-2 c d g-b e g) \int \frac {1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)}{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 (4 c e f-2 c d g-b e g) (b+2 c x)}{3 c e (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 151, normalized size = 0.92 \begin {gather*} \frac {6 b^2 e^2 (2 d g-e f+e g x)-4 b c e \left (5 d^2 g-2 d e g x+e^2 x (6 f-g x)\right )+8 c^2 \left (d^3 g+d^2 e (f-g x)+d e^2 x (2 f+g x)-2 e^3 f x^2\right )}{3 e^2 (b e-2 c d)^3 (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 [B] time = 43.04, size = 7201, normalized size = 43.64 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 9.47, size = 431, normalized size = 2.61 \begin {gather*} -\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, c^{2} e^{3} f - {\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} g\right )} x^{2} - {\left (4 \, c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} f - 2 \, {\left (2 \, c^{2} d^{3} - 5 \, b c d^{2} e + 3 \, b^{2} d e^{2}\right )} g - {\left (4 \, {\left (2 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} f - {\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} - 3 \, b^{2} e^{3}\right )} g\right )} x\right )}}{3 \, {\left (8 \, c^{5} d^{6} e^{2} - 28 \, b c^{4} d^{5} e^{3} + 38 \, b^{2} c^{3} d^{4} e^{4} - 25 \, b^{3} c^{2} d^{3} e^{5} + 8 \, b^{4} c d^{2} e^{6} - b^{5} d e^{7} + {\left (8 \, c^{5} d^{3} e^{5} - 12 \, b c^{4} d^{2} e^{6} + 6 \, b^{2} c^{3} d e^{7} - b^{3} c^{2} e^{8}\right )} x^{3} - {\left (8 \, c^{5} d^{4} e^{4} - 28 \, b c^{4} d^{3} e^{5} + 30 \, b^{2} c^{3} d^{2} e^{6} - 13 \, b^{3} c^{2} d e^{7} + 2 \, b^{4} c e^{8}\right )} x^{2} - {\left (8 \, c^{5} d^{5} e^{3} - 12 \, b c^{4} d^{4} e^{4} - 2 \, b^{2} c^{3} d^{3} e^{5} + 11 \, b^{3} c^{2} d^{2} e^{6} - 6 \, b^{4} c d e^{7} + b^{5} e^{8}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.67, size = 521, normalized size = 3.16 \begin {gather*} \frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left ({\left ({\left (\frac {2 \, {\left (4 \, c^{3} d^{2} g e^{3} - 8 \, c^{3} d f e^{4} + 4 \, b c^{2} f e^{5} - b^{2} c g e^{5}\right )} x}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}} + \frac {3 \, {\left (4 \, b c^{2} d^{2} g e^{3} - 8 \, b c^{2} d f e^{4} + 4 \, b^{2} c f e^{5} - b^{3} g e^{5}\right )}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )} x + \frac {3 \, {\left (8 \, c^{3} d^{3} f e^{2} - 4 \, b c^{2} d^{3} g e^{2} - 12 \, b c^{2} d^{2} f e^{3} + 8 \, b^{2} c d^{2} g e^{3} + 2 \, b^{2} c d f e^{4} - 3 \, b^{3} d g e^{4} + b^{3} f e^{5}\right )}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )} x + \frac {8 \, c^{3} d^{5} g + 8 \, c^{3} d^{4} f e - 24 \, b c^{2} d^{4} g e - 4 \, b c^{2} d^{3} f e^{2} + 22 \, b^{2} c d^{3} g e^{2} - 6 \, b^{2} c d^{2} f e^{3} - 6 \, b^{3} d^{2} g e^{3} + 3 \, b^{3} d f e^{4}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )}}{3 \, {\left (c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 227, normalized size = 1.38 \begin {gather*} \frac {2 \left (e x +d \right )^{2} \left (c e x +b e -c d \right ) \left (2 b c \,e^{3} g \,x^{2}+4 c^{2} d \,e^{2} g \,x^{2}-8 c^{2} e^{3} f \,x^{2}+3 b^{2} e^{3} g x +4 b c d \,e^{2} g x -12 b c \,e^{3} f x -4 c^{2} d^{2} e g x +8 c^{2} d \,e^{2} f x +6 b^{2} d \,e^{2} g -3 b^{2} e^{3} f -10 b c \,d^{2} e g +4 c^{2} d^{3} g +4 c^{2} d^{2} e f \right )}{3 \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.44, size = 795, normalized size = 4.82 \begin {gather*} -\frac {8\,c^2\,d^3\,g\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-6\,b^2\,e^3\,f\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-16\,c^2\,e^3\,f\,x^2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+12\,b^2\,d\,e^2\,g\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+8\,c^2\,d^2\,e\,f\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+6\,b^2\,e^3\,g\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+4\,b\,c\,e^3\,g\,x^2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+16\,c^2\,d\,e^2\,f\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-8\,c^2\,d^2\,e\,g\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+8\,c^2\,d\,e^2\,g\,x^2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-20\,b\,c\,d^2\,e\,g\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-24\,b\,c\,e^3\,f\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+8\,b\,c\,d\,e^2\,g\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{3\,b^5\,d\,e^7+3\,b^5\,e^8\,x-24\,b^4\,c\,d^2\,e^6-18\,b^4\,c\,d\,e^7\,x+6\,b^4\,c\,e^8\,x^2+75\,b^3\,c^2\,d^3\,e^5+33\,b^3\,c^2\,d^2\,e^6\,x-39\,b^3\,c^2\,d\,e^7\,x^2+3\,b^3\,c^2\,e^8\,x^3-114\,b^2\,c^3\,d^4\,e^4-6\,b^2\,c^3\,d^3\,e^5\,x+90\,b^2\,c^3\,d^2\,e^6\,x^2-18\,b^2\,c^3\,d\,e^7\,x^3+84\,b\,c^4\,d^5\,e^3-36\,b\,c^4\,d^4\,e^4\,x-84\,b\,c^4\,d^3\,e^5\,x^2+36\,b\,c^4\,d^2\,e^6\,x^3-24\,c^5\,d^6\,e^2+24\,c^5\,d^5\,e^3\,x+24\,c^5\,d^4\,e^4\,x^2-24\,c^5\,d^3\,e^5\,x^3} \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 ) \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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