Optimal. Leaf size=146 \[ \frac {2 (d+e x)^2 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 (d+e x) (b e g-4 c d g+2 c e f)}{3 c e^2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]
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Rubi [A] time = 0.17, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {788, 636} \begin {gather*} \frac {2 (d+e x)^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}}+\frac {2 (d+e x) (b e g-4 c d g+2 c e f)}{3 c e^2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 636
Rule 788
Rubi steps
\begin {align*} \int \frac {(d+e x)^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)^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-4 c d g+b e g) \int \frac {d+e x}{\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)^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 (2 c e f-4 c d g+b e g) (d+e x)}{3 c e^2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 100, normalized size = 0.68 \begin {gather*} \frac {2 (d+e x) \left (b e (-2 d g+3 e f+e g x)+2 c \left (d^2 g-2 d e (f+g x)+e^2 f x\right )\right )}{3 e^2 (b e-2 c d)^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 [B] time = 10.03, size = 1380, normalized size = 9.45 \begin {gather*} \frac {-2 \sqrt {c d^2-b e d-c e^2 x^2-b e^2 x} \left (-48 e^5 g x^4 c^5-24 e^5 f x^3 c^5-48 d e^4 g x^3 c^5-32 d e^4 f x^2 c^5+28 d^2 e^3 g x^2 c^5+8 d^3 e^2 f c^5-4 d^4 e g c^5+24 d^3 e^2 g x c^5+12 b e^5 g x^3 c^4+16 b e^5 f x^2 c^4-48 b d e^4 g x^2 c^4-18 b d^2 e^3 f c^4+10 b d^3 e^2 g c^4-2 b d e^4 f x c^4-50 b d^2 e^3 g x c^4+17 b^2 e^5 g x^2 c^3+13 b^2 d e^4 f c^3-8 b^2 d^2 e^3 g c^3+b^2 e^5 f x c^3+33 b^2 d e^4 g x c^3-3 b^3 e^5 f c^2+2 b^3 d e^4 g c^2-7 b^3 e^5 g x c^2\right )-2 \sqrt {-c e^2} \left (48 e^5 g x^5 c^5+24 e^5 f x^4 c^5+48 d e^4 g x^4 c^5+32 d e^4 f x^3 c^5-52 d^2 e^3 g x^3 c^5-12 d^2 e^3 f x^2 c^5-48 d^3 e^2 g x^2 c^5-4 d^4 e f c^5+8 d^5 g c^5-24 d^3 e^2 f x c^5+12 d^4 e g x c^5+12 b e^5 g x^4 c^4-4 b e^5 f x^3 c^4+96 b d e^4 g x^3 c^4+30 b d e^4 f x^2 c^4+78 b d^2 e^3 g x^2 c^4+14 b d^3 e^2 f c^4-30 b d^4 e g c^4+48 b d^2 e^3 f x c^4-36 b d^3 e^2 g x c^4-29 b^2 e^5 g x^3 c^3-12 b^2 e^5 f x^2 c^3-33 b^2 d e^4 g x^2 c^3-18 b^2 d^2 e^3 f c^3+43 b^2 d^3 e^2 g c^3-30 b^2 d e^4 f x c^3+39 b^2 d^2 e^3 g x c^3+3 b^3 e^5 g x^2 c^2+10 b^3 d e^4 f c^2-29 b^3 d^2 e^3 g c^2+6 b^3 e^5 f x c^2-18 b^3 d e^4 g x c^2-2 b^4 e^5 f c+9 b^4 d e^4 g c+3 b^4 e^5 g x c-b^5 e^5 g\right )}{3 c^2 \sqrt {-c e^2} \sqrt {c d^2-b e d-c e^2 x^2-b e^2 x} \left (32 e^4 x^5 c^5-56 d^2 e^2 x^3 c^5+24 d^4 x c^5+32 b e^4 x^4 c^4+80 b d e^3 x^3 c^4-24 b d^2 e^2 x^2 c^4-72 b d^3 e x c^4-18 b^2 e^4 x^3 c^3+36 b^2 d e^3 x^2 c^3+78 b^2 d^2 e^2 x c^3-12 b^3 e^4 x^2 c^2-36 b^3 d e^3 x c^2+6 b^4 e^4 x c\right ) e^3+3 c^2 \left (-8 d^6 c^6+32 e^6 x^6 c^6-72 d^2 e^4 x^4 c^6+48 d^4 e^2 x^2 c^6+48 b e^6 x^5 c^5+96 b d e^5 x^4 c^5-60 b d^2 e^4 x^3 c^5-132 b d^3 e^3 x^2 c^5+36 b d^5 e c^5+12 b d^4 e^2 x c^5-6 b^2 e^6 x^4 c^4+84 b^2 d e^5 x^3 c^4-66 b^2 d^4 e^2 c^4+120 b^2 d^2 e^4 x^2 c^4-36 b^2 d^3 e^3 x c^4+63 b^3 d^3 e^3 c^3-23 b^3 e^6 x^3 c^3-39 b^3 d e^5 x^2 c^3+39 b^3 d^2 e^4 x c^3-33 b^4 d^2 e^4 c^2+3 b^4 e^6 x^2 c^2-18 b^4 d e^5 x c^2+9 b^5 d e^5 c+3 b^5 e^6 x c-b^6 e^6\right ) e^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.46, size = 227, normalized size = 1.55 \begin {gather*} \frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (4 \, c d e - 3 \, b e^{2}\right )} f - 2 \, {\left (c d^{2} - b d e\right )} g - {\left (2 \, c e^{2} f - {\left (4 \, c d e - b e^{2}\right )} g\right )} x\right )}}{3 \, {\left (4 \, c^{4} d^{4} e^{2} - 12 \, b c^{3} d^{3} e^{3} + 13 \, b^{2} c^{2} d^{2} e^{4} - 6 \, b^{3} c d e^{5} + b^{4} e^{6} + {\left (4 \, c^{4} d^{2} e^{4} - 4 \, b c^{3} d e^{5} + b^{2} c^{2} e^{6}\right )} x^{2} - 2 \, {\left (4 \, c^{4} d^{3} e^{3} - 8 \, b c^{3} d^{2} e^{4} + 5 \, b^{2} c^{2} d e^{5} - b^{3} c e^{6}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.73, size = 587, normalized size = 4.02 \begin {gather*} \frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left ({\left ({\left (\frac {{\left (16 \, c^{3} d^{3} g e^{3} - 8 \, c^{3} d^{2} f e^{4} - 20 \, b c^{2} d^{2} g e^{4} + 8 \, b c^{2} d f e^{5} + 8 \, b^{2} c d g e^{5} - 2 \, b^{2} c f e^{6} - b^{3} g e^{6}\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 (8 \, c^{3} d^{4} g e^{2} - 8 \, b c^{2} d^{3} g e^{3} - 4 \, b c^{2} d^{2} f e^{4} + 2 \, b^{2} c d^{2} g e^{4} + 4 \, b^{2} c d f e^{5} - b^{3} f e^{6}\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^{4} f e^{2} + 4 \, b c^{2} d^{4} g e^{2} - 16 \, b c^{2} d^{3} f e^{3} - 4 \, b^{2} c d^{3} g e^{3} + 10 \, b^{2} c d^{2} f e^{4} + b^{3} d^{2} g e^{4} - 2 \, b^{3} d 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^{6} g - 16 \, c^{3} d^{5} f e - 16 \, b c^{2} d^{5} g e + 28 \, b c^{2} d^{4} f e^{2} + 10 \, b^{2} c d^{4} g e^{2} - 16 \, b^{2} c d^{3} f e^{3} - 2 \, b^{3} d^{3} g e^{3} + 3 \, b^{3} d^{2} 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 = 0.05, size = 128, normalized size = 0.88 \begin {gather*} -\frac {2 \left (e x +d \right )^{3} \left (c e x +b e -c d \right ) \left (-b \,e^{2} g x +4 c d e g x -2 c \,e^{2} f x +2 b d e g -3 b \,e^{2} f -2 c \,d^{2} g +4 c d e f \right )}{3 \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\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 = 3.30, size = 107, normalized size = 0.73 \begin {gather*} -\frac {2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (3\,b\,e^2\,f+2\,c\,d^2\,g+b\,e^2\,g\,x+2\,c\,e^2\,f\,x-2\,b\,d\,e\,g-4\,c\,d\,e\,f-4\,c\,d\,e\,g\,x\right )}{3\,e^2\,{\left (b\,e-2\,c\,d\right )}^2\,{\left (b\,e-c\,d+c\,e\,x\right )}^2} \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 )^{2} \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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