3.9.15 \(\int \frac {(a+b x+c x^2)^2}{(d+e x) (f+g x)} \, dx\) [815]

Optimal. Leaf size=184 \[ \frac {\left (b^2 e^2 g^2-2 c e g (b e f+b d g-a e g)+c^2 \left (e^2 f^2+d e f g+d^2 g^2\right )\right ) x}{e^3 g^3}-\frac {c (c e f+c d g-2 b e g) x^2}{2 e^2 g^2}+\frac {c^2 x^3}{3 e g}+\frac {\left (c d^2-b d e+a e^2\right )^2 \log (d+e x)}{e^4 (e f-d g)}-\frac {\left (c f^2-b f g+a g^2\right )^2 \log (f+g x)}{g^4 (e f-d g)} \]

[Out]

(b^2*e^2*g^2-2*c*e*g*(-a*e*g+b*d*g+b*e*f)+c^2*(d^2*g^2+d*e*f*g+e^2*f^2))*x/e^3/g^3-1/2*c*(-2*b*e*g+c*d*g+c*e*f
)*x^2/e^2/g^2+1/3*c^2*x^3/e/g+(a*e^2-b*d*e+c*d^2)^2*ln(e*x+d)/e^4/(-d*g+e*f)-(a*g^2-b*f*g+c*f^2)^2*ln(g*x+f)/g
^4/(-d*g+e*f)

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Rubi [A]
time = 0.19, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {907} \begin {gather*} \frac {x \left (-2 c e g (-a e g+b d g+b e f)+b^2 e^2 g^2+c^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{e^3 g^3}+\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^4 (e f-d g)}-\frac {\log (f+g x) \left (a g^2-b f g+c f^2\right )^2}{g^4 (e f-d g)}-\frac {c x^2 (-2 b e g+c d g+c e f)}{2 e^2 g^2}+\frac {c^2 x^3}{3 e g} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^2/((d + e*x)*(f + g*x)),x]

[Out]

((b^2*e^2*g^2 - 2*c*e*g*(b*e*f + b*d*g - a*e*g) + c^2*(e^2*f^2 + d*e*f*g + d^2*g^2))*x)/(e^3*g^3) - (c*(c*e*f
+ c*d*g - 2*b*e*g)*x^2)/(2*e^2*g^2) + (c^2*x^3)/(3*e*g) + ((c*d^2 - b*d*e + a*e^2)^2*Log[d + e*x])/(e^4*(e*f -
d*g)) - ((c*f^2 - b*f*g + a*g^2)^2*Log[f + g*x])/(g^4*(e*f - d*g))

Rule 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x) (f+g x)} \, dx &=\int \left (\frac {b^2 e^2 g^2-2 c e g (b e f+b d g-a e g)+c^2 \left (e^2 f^2+d e f g+d^2 g^2\right )}{e^3 g^3}-\frac {c (c e f+c d g-2 b e g) x}{e^2 g^2}+\frac {c^2 x^2}{e g}+\frac {\left (c d^2-b d e+a e^2\right )^2}{e^3 (e f-d g) (d+e x)}+\frac {\left (c f^2-b f g+a g^2\right )^2}{g^3 (-e f+d g) (f+g x)}\right ) \, dx\\ &=\frac {\left (b^2 e^2 g^2-2 c e g (b e f+b d g-a e g)+c^2 \left (e^2 f^2+d e f g+d^2 g^2\right )\right ) x}{e^3 g^3}-\frac {c (c e f+c d g-2 b e g) x^2}{2 e^2 g^2}+\frac {c^2 x^3}{3 e g}+\frac {\left (c d^2-b d e+a e^2\right )^2 \log (d+e x)}{e^4 (e f-d g)}-\frac {\left (c f^2-b f g+a g^2\right )^2 \log (f+g x)}{g^4 (e f-d g)}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 177, normalized size = 0.96 \begin {gather*} -\frac {e g (-e f+d g) x \left (6 b^2 e^2 g^2+6 c e g (2 a e g+b (-2 e f-2 d g+e g x))+c^2 \left (6 d^2 g^2-3 d e g (-2 f+g x)+e^2 \left (6 f^2-3 f g x+2 g^2 x^2\right )\right )\right )-6 \left (c d^2+e (-b d+a e)\right )^2 g^4 \log (d+e x)+6 e^4 \left (c f^2+g (-b f+a g)\right )^2 \log (f+g x)}{6 e^4 g^4 (e f-d g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^2/((d + e*x)*(f + g*x)),x]

[Out]

-1/6*(e*g*(-(e*f) + d*g)*x*(6*b^2*e^2*g^2 + 6*c*e*g*(2*a*e*g + b*(-2*e*f - 2*d*g + e*g*x)) + c^2*(6*d^2*g^2 -
3*d*e*g*(-2*f + g*x) + e^2*(6*f^2 - 3*f*g*x + 2*g^2*x^2))) - 6*(c*d^2 + e*(-(b*d) + a*e))^2*g^4*Log[d + e*x] +
6*e^4*(c*f^2 + g*(-(b*f) + a*g))^2*Log[f + g*x])/(e^4*g^4*(e*f - d*g))

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Maple [A]
time = 0.15, size = 280, normalized size = 1.52

method	result	size
norman	\(\frac {\left (2 a c \,e^{2} g^{2}+b^{2} e^{2} g^{2}-2 b c d e \,g^{2}-2 b c \,e^{2} f g +c^{2} d^{2} g^{2}+c^{2} d e f g +c^{2} e^{2} f^{2}\right ) x}{e^{3} g^{3}}+\frac {c^{2} x^{3}}{3 e g}+\frac {c \left (2 b e g -d g c -c e f \right ) x^{2}}{2 e^{2} g^{2}}+\frac {\left (a^{2} g^{4}-2 a b f \,g^{3}+2 a c \,f^{2} g^{2}+b^{2} f^{2} g^{2}-2 b c \,f^{3} g +c^{2} f^{4}\right ) \ln \left (g x +f \right )}{g^{4} \left (d g -e f \right )}-\frac {\left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) \ln \left (e x +d \right )}{\left (d g -e f \right ) e^{4}}\)	\(262\)
default	\(\frac {\frac {1}{3} c^{2} x^{3} e^{2} g^{2}+b c \,e^{2} g^{2} x^{2}-\frac {1}{2} c^{2} d e \,g^{2} x^{2}-\frac {1}{2} c^{2} e^{2} f g \,x^{2}+2 a c \,e^{2} g^{2} x +b^{2} e^{2} g^{2} x -2 b c d e \,g^{2} x -2 b c \,e^{2} f g x +c^{2} d^{2} g^{2} x +c^{2} d e f g x +c^{2} e^{2} f^{2} x}{e^{3} g^{3}}+\frac {\left (-a^{2} e^{4}+2 a b d \,e^{3}-2 a c \,d^{2} e^{2}-b^{2} d^{2} e^{2}+2 b c \,d^{3} e -c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{4} \left (d g -e f \right )}+\frac {\left (a^{2} g^{4}-2 a b f \,g^{3}+2 a c \,f^{2} g^{2}+b^{2} f^{2} g^{2}-2 b c \,f^{3} g +c^{2} f^{4}\right ) \ln \left (g x +f \right )}{g^{4} \left (d g -e f \right )}\)	\(280\)
risch	\(\frac {c^{2} x^{3}}{3 e g}+\frac {b c \,x^{2}}{e g}-\frac {c^{2} d \,x^{2}}{2 e^{2} g}-\frac {c^{2} f \,x^{2}}{2 e \,g^{2}}+\frac {2 a c x}{e g}+\frac {b^{2} x}{e g}-\frac {2 b c d x}{e^{2} g}-\frac {2 b c f x}{e \,g^{2}}+\frac {c^{2} d^{2} x}{e^{3} g}+\frac {c^{2} d f x}{e^{2} g^{2}}+\frac {c^{2} f^{2} x}{e \,g^{3}}+\frac {\ln \left (-g x -f \right ) a^{2}}{d g -e f}-\frac {2 \ln \left (-g x -f \right ) a b f}{g \left (d g -e f \right )}+\frac {2 \ln \left (-g x -f \right ) a c \,f^{2}}{g^{2} \left (d g -e f \right )}+\frac {\ln \left (-g x -f \right ) b^{2} f^{2}}{g^{2} \left (d g -e f \right )}-\frac {2 \ln \left (-g x -f \right ) b c \,f^{3}}{g^{3} \left (d g -e f \right )}+\frac {\ln \left (-g x -f \right ) c^{2} f^{4}}{g^{4} \left (d g -e f \right )}-\frac {\ln \left (e x +d \right ) a^{2}}{d g -e f}+\frac {2 \ln \left (e x +d \right ) a b d}{\left (d g -e f \right ) e}-\frac {2 \ln \left (e x +d \right ) a c \,d^{2}}{\left (d g -e f \right ) e^{2}}-\frac {\ln \left (e x +d \right ) b^{2} d^{2}}{\left (d g -e f \right ) e^{2}}+\frac {2 \ln \left (e x +d \right ) b c \,d^{3}}{\left (d g -e f \right ) e^{3}}-\frac {\ln \left (e x +d \right ) c^{2} d^{4}}{\left (d g -e f \right ) e^{4}}\)	\(462\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^2/(e*x+d)/(g*x+f),x,method=_RETURNVERBOSE)

[Out]

1/e^3/g^3*(1/3*c^2*x^3*e^2*g^2+b*c*e^2*g^2*x^2-1/2*c^2*d*e*g^2*x^2-1/2*c^2*e^2*f*g*x^2+2*a*c*e^2*g^2*x+b^2*e^2
*g^2*x-2*b*c*d*e*g^2*x-2*b*c*e^2*f*g*x+c^2*d^2*g^2*x+c^2*d*e*f*g*x+c^2*e^2*f^2*x)+(-a^2*e^4+2*a*b*d*e^3-2*a*c*
d^2*e^2-b^2*d^2*e^2+2*b*c*d^3*e-c^2*d^4)/e^4/(d*g-e*f)*ln(e*x+d)+1/g^4*(a^2*g^4-2*a*b*f*g^3+2*a*c*f^2*g^2+b^2*
f^2*g^2-2*b*c*f^3*g+c^2*f^4)/(d*g-e*f)*ln(g*x+f)

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Maxima [A]
time = 0.27, size = 253, normalized size = 1.38 \begin {gather*} \frac {{\left (c^{2} f^{4} - 2 \, b c f^{3} g - 2 \, a b f g^{3} + a^{2} g^{4} + {\left (b^{2} + 2 \, a c\right )} f^{2} g^{2}\right )} \log \left (g x + f\right )}{d g^{5} - f g^{4} e} - \frac {{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d^{2} + a^{2} e^{4}\right )} \log \left (x e + d\right )}{d g e^{4} - f e^{5}} + \frac {{\left (2 \, c^{2} g^{2} x^{3} e^{2} - 3 \, {\left (c^{2} f g e^{2} + {\left (c^{2} d e - 2 \, b c e^{2}\right )} g^{2}\right )} x^{2} + 6 \, {\left (c^{2} f^{2} e^{2} + {\left (c^{2} d e - 2 \, b c e^{2}\right )} f g + {\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} g^{2}\right )} x\right )} e^{\left (-3\right )}}{6 \, g^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)/(g*x+f),x, algorithm="maxima")

[Out]

(c^2*f^4 - 2*b*c*f^3*g - 2*a*b*f*g^3 + a^2*g^4 + (b^2 + 2*a*c)*f^2*g^2)*log(g*x + f)/(d*g^5 - f*g^4*e) - (c^2*
d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + (b^2*e^2 + 2*a*c*e^2)*d^2 + a^2*e^4)*log(x*e + d)/(d*g*e^4 - f*e^5) + 1/6*(2
*c^2*g^2*x^3*e^2 - 3*(c^2*f*g*e^2 + (c^2*d*e - 2*b*c*e^2)*g^2)*x^2 + 6*(c^2*f^2*e^2 + (c^2*d*e - 2*b*c*e^2)*f*
g + (c^2*d^2 - 2*b*c*d*e + b^2*e^2 + 2*a*c*e^2)*g^2)*x)*e^(-3)/g^3

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Fricas [A]
time = 1.38, size = 306, normalized size = 1.66 \begin {gather*} \frac {6 \, c^{2} d^{3} g^{4} x e + 6 \, {\left (c^{2} f^{4} - 2 \, b c f^{3} g - 2 \, a b f g^{3} + a^{2} g^{4} + {\left (b^{2} + 2 \, a c\right )} f^{2} g^{2}\right )} e^{4} \log \left (g x + f\right ) - {\left (2 \, c^{2} f g^{3} x^{3} - 3 \, {\left (c^{2} f^{2} g^{2} - 2 \, b c f g^{3}\right )} x^{2} + 6 \, {\left (c^{2} f^{3} g - 2 \, b c f^{2} g^{2} + {\left (b^{2} + 2 \, a c\right )} f g^{3}\right )} x\right )} e^{4} + 2 \, {\left (c^{2} d g^{4} x^{3} + 3 \, b c d g^{4} x^{2} + 3 \, {\left (b^{2} + 2 \, a c\right )} d g^{4} x\right )} e^{3} - 3 \, {\left (c^{2} d^{2} g^{4} x^{2} + 4 \, b c d^{2} g^{4} x\right )} e^{2} - 6 \, {\left (c^{2} d^{4} g^{4} - 2 \, b c d^{3} g^{4} e - 2 \, a b d g^{4} e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} g^{4} e^{2} + a^{2} g^{4} e^{4}\right )} \log \left (x e + d\right )}{6 \, {\left (d g^{5} e^{4} - f g^{4} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)/(g*x+f),x, algorithm="fricas")

[Out]

1/6*(6*c^2*d^3*g^4*x*e + 6*(c^2*f^4 - 2*b*c*f^3*g - 2*a*b*f*g^3 + a^2*g^4 + (b^2 + 2*a*c)*f^2*g^2)*e^4*log(g*x
+ f) - (2*c^2*f*g^3*x^3 - 3*(c^2*f^2*g^2 - 2*b*c*f*g^3)*x^2 + 6*(c^2*f^3*g - 2*b*c*f^2*g^2 + (b^2 + 2*a*c)*f*
g^3)*x)*e^4 + 2*(c^2*d*g^4*x^3 + 3*b*c*d*g^4*x^2 + 3*(b^2 + 2*a*c)*d*g^4*x)*e^3 - 3*(c^2*d^2*g^4*x^2 + 4*b*c*d
^2*g^4*x)*e^2 - 6*(c^2*d^4*g^4 - 2*b*c*d^3*g^4*e - 2*a*b*d*g^4*e^3 + (b^2 + 2*a*c)*d^2*g^4*e^2 + a^2*g^4*e^4)*
log(x*e + d))/(d*g^5*e^4 - f*g^4*e^5)

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Sympy [F(-1)] Timed out
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.

[In]

integrate((c*x**2+b*x+a)**2/(e*x+d)/(g*x+f),x)

[Out]

Timed out

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Giac [A]
time = 5.77, size = 281, normalized size = 1.53 \begin {gather*} \frac {{\left (c^{2} f^{4} - 2 \, b c f^{3} g + b^{2} f^{2} g^{2} + 2 \, a c f^{2} g^{2} - 2 \, a b f g^{3} + a^{2} g^{4}\right )} \log \left ({\left | g x + f \right |}\right )}{d g^{5} - f g^{4} e} - \frac {{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{d g e^{4} - f e^{5}} + \frac {{\left (2 \, c^{2} g^{2} x^{3} e^{2} - 3 \, c^{2} d g^{2} x^{2} e + 6 \, c^{2} d^{2} g^{2} x - 3 \, c^{2} f g x^{2} e^{2} + 6 \, b c g^{2} x^{2} e^{2} + 6 \, c^{2} d f g x e - 12 \, b c d g^{2} x e + 6 \, c^{2} f^{2} x e^{2} - 12 \, b c f g x e^{2} + 6 \, b^{2} g^{2} x e^{2} + 12 \, a c g^{2} x e^{2}\right )} e^{\left (-3\right )}}{6 \, g^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)/(g*x+f),x, algorithm="giac")

[Out]

(c^2*f^4 - 2*b*c*f^3*g + b^2*f^2*g^2 + 2*a*c*f^2*g^2 - 2*a*b*f*g^3 + a^2*g^4)*log(abs(g*x + f))/(d*g^5 - f*g^4
*e) - (c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*log(abs(x*e + d))/(d*g*e^4
- f*e^5) + 1/6*(2*c^2*g^2*x^3*e^2 - 3*c^2*d*g^2*x^2*e + 6*c^2*d^2*g^2*x - 3*c^2*f*g*x^2*e^2 + 6*b*c*g^2*x^2*e
^2 + 6*c^2*d*f*g*x*e - 12*b*c*d*g^2*x*e + 6*c^2*f^2*x*e^2 - 12*b*c*f*g*x*e^2 + 6*b^2*g^2*x*e^2 + 12*a*c*g^2*x*
e^2)*e^(-3)/g^3

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Mupad [B]
time = 3.51, size = 266, normalized size = 1.45 \begin {gather*} x\,\left (\frac {b^2+2\,a\,c}{e\,g}+\frac {\left (\frac {c^2\,\left (d\,g+e\,f\right )}{e^2\,g^2}-\frac {2\,b\,c}{e\,g}\right )\,\left (d\,g+e\,f\right )}{e\,g}-\frac {c^2\,d\,f}{e^2\,g^2}\right )-x^2\,\left (\frac {c^2\,\left (d\,g+e\,f\right )}{2\,e^2\,g^2}-\frac {b\,c}{e\,g}\right )+\frac {\ln \left (d+e\,x\right )\,\left (e^2\,\left (b^2\,d^2+2\,a\,c\,d^2\right )+a^2\,e^4+c^2\,d^4-2\,a\,b\,d\,e^3-2\,b\,c\,d^3\,e\right )}{e^5\,f-d\,e^4\,g}+\frac {\ln \left (f+g\,x\right )\,\left (g^2\,\left (b^2\,f^2+2\,a\,c\,f^2\right )+a^2\,g^4+c^2\,f^4-2\,a\,b\,f\,g^3-2\,b\,c\,f^3\,g\right )}{d\,g^5-e\,f\,g^4}+\frac {c^2\,x^3}{3\,e\,g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^2/((f + g*x)*(d + e*x)),x)

[Out]

x*((2*a*c + b^2)/(e*g) + (((c^2*(d*g + e*f))/(e^2*g^2) - (2*b*c)/(e*g))*(d*g + e*f))/(e*g) - (c^2*d*f)/(e^2*g^
2)) - x^2*((c^2*(d*g + e*f))/(2*e^2*g^2) - (b*c)/(e*g)) + (log(d + e*x)*(e^2*(b^2*d^2 + 2*a*c*d^2) + a^2*e^4 +
c^2*d^4 - 2*a*b*d*e^3 - 2*b*c*d^3*e))/(e^5*f - d*e^4*g) + (log(f + g*x)*(g^2*(b^2*f^2 + 2*a*c*f^2) + a^2*g^4
+ c^2*f^4 - 2*a*b*f*g^3 - 2*b*c*f^3*g))/(d*g^5 - e*f*g^4) + (c^2*x^3)/(3*e*g)

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