3.1.17 \(\int \frac {g+h x}{(a+b x+c x^2) (a d+b d x+c d x^2)^2} \, dx\) [17]

Optimal. Leaf size=140 \[ -\frac {b g-2 a h+(2 c g-b h) x}{2 \left (b^2-4 a c\right ) d^2 \left (a+b x+c x^2\right )^2}+\frac {3 (2 c g-b h) (b+2 c x)}{2 \left (b^2-4 a c\right )^2 d^2 \left (a+b x+c x^2\right )}-\frac {6 c (2 c g-b h) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^2} \]

[Out]

1/2*(-b*g+2*a*h-(-b*h+2*c*g)*x)/(-4*a*c+b^2)/d^2/(c*x^2+b*x+a)^2+3/2*(-b*h+2*c*g)*(2*c*x+b)/(-4*a*c+b^2)^2/d^2
/(c*x^2+b*x+a)-6*c*(-b*h+2*c*g)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(5/2)/d^2

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Rubi [A]
time = 0.09, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {1012, 652, 628, 632, 212} \begin {gather*} \frac {3 (b+2 c x) (2 c g-b h)}{2 d^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {-2 a h+x (2 c g-b h)+b g}{2 d^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {6 c (2 c g-b h) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b*g - 2*a*h + (2*c*g - b*h)*x)/((b^2 - 4*a*c)*d^2*(a + b*x + c*x^2)^2) + (3*(2*c*g - b*h)*(b + 2*c*x))/(
2*(b^2 - 4*a*c)^2*d^2*(a + b*x + c*x^2)) - (6*c*(2*c*g - b*h)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 -
4*a*c)^(5/2)*d^2)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 1012

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_)
, x_Symbol] :> Dist[(c/f)^p, Int[(g + h*x)^m*(d + e*x + f*x^2)^(p + q), x], x] /; FreeQ[{a, b, c, d, e, f, g,
h, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && ( !IntegerQ[q] || Le
afCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])

Rubi steps

\begin {align*} \int \frac {g+h x}{\left (a+b x+c x^2\right ) \left (a d+b d x+c d x^2\right )^2} \, dx &=\frac {\int \frac {g+h x}{\left (a+b x+c x^2\right )^3} \, dx}{d^2}\\ &=-\frac {b g-2 a h+(2 c g-b h) x}{2 \left (b^2-4 a c\right ) d^2 \left (a+b x+c x^2\right )^2}-\frac {(3 (2 c g-b h)) \int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right ) d^2}\\ &=-\frac {b g-2 a h+(2 c g-b h) x}{2 \left (b^2-4 a c\right ) d^2 \left (a+b x+c x^2\right )^2}+\frac {3 (2 c g-b h) (b+2 c x)}{2 \left (b^2-4 a c\right )^2 d^2 \left (a+b x+c x^2\right )}+\frac {(3 c (2 c g-b h)) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 d^2}\\ &=-\frac {b g-2 a h+(2 c g-b h) x}{2 \left (b^2-4 a c\right ) d^2 \left (a+b x+c x^2\right )^2}+\frac {3 (2 c g-b h) (b+2 c x)}{2 \left (b^2-4 a c\right )^2 d^2 \left (a+b x+c x^2\right )}-\frac {(6 c (2 c g-b h)) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2 d^2}\\ &=-\frac {b g-2 a h+(2 c g-b h) x}{2 \left (b^2-4 a c\right ) d^2 \left (a+b x+c x^2\right )^2}+\frac {3 (2 c g-b h) (b+2 c x)}{2 \left (b^2-4 a c\right )^2 d^2 \left (a+b x+c x^2\right )}-\frac {6 c (2 c g-b h) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^2}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 131, normalized size = 0.94 \begin {gather*} \frac {\frac {\left (b^2-4 a c\right ) (-b g+2 a h-2 c g x+b h x)}{(a+x (b+c x))^2}+\frac {3 (2 c g-b h) (b+2 c x)}{a+x (b+c x)}-\frac {12 c (-2 c g+b h) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{2 \left (b^2-4 a c\right )^2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((b^2 - 4*a*c)*(-(b*g) + 2*a*h - 2*c*g*x + b*h*x))/(a + x*(b + c*x))^2 + (3*(2*c*g - b*h)*(b + 2*c*x))/(a + x
*(b + c*x)) - (12*c*(-2*c*g + b*h)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c
)^2*d^2)

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Maple [A]
time = 0.18, size = 141, normalized size = 1.01

method	result	size
default	\(\frac {\frac {b g -2 a h +\left (-b h +2 c g \right ) x}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 \left (-b h +2 c g \right ) \left (\frac {2 c x +b}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )}+\frac {4 c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (4 a c -b^{2}\right )}}{d^{2}}\)	\(141\)
risch	\(\frac {-\frac {3 c^{2} \left (b h -2 c g \right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {9 b c \left (b h -2 c g \right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (5 a b c h -10 a \,c^{2} g +b^{3} h -2 b^{2} c g \right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {8 a^{2} c h +a \,b^{2} h -10 a b c g +b^{3} g}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2} d^{2}}-\frac {3 c \ln \left (\left (32 a^{2} c^{3}-16 a \,b^{2} c^{2}+2 b^{4} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) b h}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}}+\frac {6 c^{2} \ln \left (\left (32 a^{2} c^{3}-16 a \,b^{2} c^{2}+2 b^{4} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) g}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}}+\frac {3 c \ln \left (\left (-32 a^{2} c^{3}+16 a \,b^{2} c^{2}-2 b^{4} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) b h}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}}-\frac {6 c^{2} \ln \left (\left (-32 a^{2} c^{3}+16 a \,b^{2} c^{2}-2 b^{4} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) g}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}}\)	\(504\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^2*(1/2*(b*g-2*a*h+(-b*h+2*c*g)*x)/(4*a*c-b^2)/(c*x^2+b*x+a)^2+3/2*(-b*h+2*c*g)/(4*a*c-b^2)*((2*c*x+b)/(4*a
*c-b^2)/(c*x^2+b*x+a)+4*c/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))

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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.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (132) = 264\).
time = 0.53, size = 1150, normalized size = 8.21 \begin {gather*} \left [\frac {6 \, {\left (2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} g - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} h\right )} x^{3} + 9 \, {\left (2 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} g - {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} h\right )} x^{2} - 6 \, {\left (2 \, a^{2} c^{2} g - a^{2} b c h + {\left (2 \, c^{4} g - b c^{3} h\right )} x^{4} + 2 \, {\left (2 \, b c^{3} g - b^{2} c^{2} h\right )} x^{3} + {\left (2 \, {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} g - {\left (b^{3} c + 2 \, a b c^{2}\right )} h\right )} x^{2} + 2 \, {\left (2 \, a b c^{2} g - a b^{2} c h\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - {\left (b^{5} - 14 \, a b^{3} c + 40 \, a^{2} b c^{2}\right )} g - {\left (a b^{4} + 4 \, a^{2} b^{2} c - 32 \, a^{3} c^{2}\right )} h + 2 \, {\left (2 \, {\left (b^{4} c + a b^{2} c^{2} - 20 \, a^{2} c^{3}\right )} g - {\left (b^{5} + a b^{3} c - 20 \, a^{2} b c^{2}\right )} h\right )} x}{2 \, {\left ({\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} x^{4} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{2} x^{3} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} d^{2} x^{2} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} d^{2} x + {\left (a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3}\right )} d^{2}\right )}}, \frac {6 \, {\left (2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} g - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} h\right )} x^{3} + 9 \, {\left (2 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} g - {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} h\right )} x^{2} - 12 \, {\left (2 \, a^{2} c^{2} g - a^{2} b c h + {\left (2 \, c^{4} g - b c^{3} h\right )} x^{4} + 2 \, {\left (2 \, b c^{3} g - b^{2} c^{2} h\right )} x^{3} + {\left (2 \, {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} g - {\left (b^{3} c + 2 \, a b c^{2}\right )} h\right )} x^{2} + 2 \, {\left (2 \, a b c^{2} g - a b^{2} c h\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - {\left (b^{5} - 14 \, a b^{3} c + 40 \, a^{2} b c^{2}\right )} g - {\left (a b^{4} + 4 \, a^{2} b^{2} c - 32 \, a^{3} c^{2}\right )} h + 2 \, {\left (2 \, {\left (b^{4} c + a b^{2} c^{2} - 20 \, a^{2} c^{3}\right )} g - {\left (b^{5} + a b^{3} c - 20 \, a^{2} b c^{2}\right )} h\right )} x}{2 \, {\left ({\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} x^{4} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{2} x^{3} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} d^{2} x^{2} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} d^{2} x + {\left (a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3}\right )} d^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(6*(2*(b^2*c^3 - 4*a*c^4)*g - (b^3*c^2 - 4*a*b*c^3)*h)*x^3 + 9*(2*(b^3*c^2 - 4*a*b*c^3)*g - (b^4*c - 4*a*
b^2*c^2)*h)*x^2 - 6*(2*a^2*c^2*g - a^2*b*c*h + (2*c^4*g - b*c^3*h)*x^4 + 2*(2*b*c^3*g - b^2*c^2*h)*x^3 + (2*(b
^2*c^2 + 2*a*c^3)*g - (b^3*c + 2*a*b*c^2)*h)*x^2 + 2*(2*a*b*c^2*g - a*b^2*c*h)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2
*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) - (b^5 - 14*a*b^3*c + 40*a^2*
b*c^2)*g - (a*b^4 + 4*a^2*b^2*c - 32*a^3*c^2)*h + 2*(2*(b^4*c + a*b^2*c^2 - 20*a^2*c^3)*g - (b^5 + a*b^3*c - 2
0*a^2*b*c^2)*h)*x)/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*x^4 + 2*(b^7*c - 12*a*b^5*c^2 +
48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2*x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d
^2*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2*x + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2
*c^2 - 64*a^5*c^3)*d^2), 1/2*(6*(2*(b^2*c^3 - 4*a*c^4)*g - (b^3*c^2 - 4*a*b*c^3)*h)*x^3 + 9*(2*(b^3*c^2 - 4*a*
b*c^3)*g - (b^4*c - 4*a*b^2*c^2)*h)*x^2 - 12*(2*a^2*c^2*g - a^2*b*c*h + (2*c^4*g - b*c^3*h)*x^4 + 2*(2*b*c^3*g
- b^2*c^2*h)*x^3 + (2*(b^2*c^2 + 2*a*c^3)*g - (b^3*c + 2*a*b*c^2)*h)*x^2 + 2*(2*a*b*c^2*g - a*b^2*c*h)*x)*sqr
t(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (b^5 - 14*a*b^3*c + 40*a^2*b*c^2)*g -
(a*b^4 + 4*a^2*b^2*c - 32*a^3*c^2)*h + 2*(2*(b^4*c + a*b^2*c^2 - 20*a^2*c^3)*g - (b^5 + a*b^3*c - 20*a^2*b*c^2
)*h)*x)/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3
*c^3 - 64*a^3*b*c^4)*d^2*x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*x^2 + 2*
(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2*x + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a
^5*c^3)*d^2)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (133) = 266\).
time = 1.21, size = 709, normalized size = 5.06 \begin {gather*} \frac {3 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) \log {\left (x + \frac {- 192 a^{3} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) + 144 a^{2} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) - 36 a b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) + 3 b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) + 3 b^{2} c h - 6 b c^{2} g}{6 b c^{2} h - 12 c^{3} g} \right )}}{d^{2}} - \frac {3 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) \log {\left (x + \frac {192 a^{3} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) - 144 a^{2} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) + 36 a b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) - 3 b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b h - 2 c g\right ) + 3 b^{2} c h - 6 b c^{2} g}{6 b c^{2} h - 12 c^{3} g} \right )}}{d^{2}} + \frac {- 8 a^{2} c h - a b^{2} h + 10 a b c g - b^{3} g + x^{3} \left (- 6 b c^{2} h + 12 c^{3} g\right ) + x^{2} \left (- 9 b^{2} c h + 18 b c^{2} g\right ) + x \left (- 10 a b c h + 20 a c^{2} g - 2 b^{3} h + 4 b^{2} c g\right )}{32 a^{4} c^{2} d^{2} - 16 a^{3} b^{2} c d^{2} + 2 a^{2} b^{4} d^{2} + x^{4} \cdot \left (32 a^{2} c^{4} d^{2} - 16 a b^{2} c^{3} d^{2} + 2 b^{4} c^{2} d^{2}\right ) + x^{3} \cdot \left (64 a^{2} b c^{3} d^{2} - 32 a b^{3} c^{2} d^{2} + 4 b^{5} c d^{2}\right ) + x^{2} \cdot \left (64 a^{3} c^{3} d^{2} - 12 a b^{4} c d^{2} + 2 b^{6} d^{2}\right ) + x \left (64 a^{3} b c^{2} d^{2} - 32 a^{2} b^{3} c d^{2} + 4 a b^{5} d^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*c*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g)*log(x + (-192*a**3*c**4*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g)
+ 144*a**2*b**2*c**3*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g) - 36*a*b**4*c**2*sqrt(-1/(4*a*c - b**2)**5)*(b*h
- 2*c*g) + 3*b**6*c*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g) + 3*b**2*c*h - 6*b*c**2*g)/(6*b*c**2*h - 12*c**3
*g))/d**2 - 3*c*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g)*log(x + (192*a**3*c**4*sqrt(-1/(4*a*c - b**2)**5)*(b*
h - 2*c*g) - 144*a**2*b**2*c**3*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g) + 36*a*b**4*c**2*sqrt(-1/(4*a*c - b**
2)**5)*(b*h - 2*c*g) - 3*b**6*c*sqrt(-1/(4*a*c - b**2)**5)*(b*h - 2*c*g) + 3*b**2*c*h - 6*b*c**2*g)/(6*b*c**2*
h - 12*c**3*g))/d**2 + (-8*a**2*c*h - a*b**2*h + 10*a*b*c*g - b**3*g + x**3*(-6*b*c**2*h + 12*c**3*g) + x**2*(
-9*b**2*c*h + 18*b*c**2*g) + x*(-10*a*b*c*h + 20*a*c**2*g - 2*b**3*h + 4*b**2*c*g))/(32*a**4*c**2*d**2 - 16*a*
*3*b**2*c*d**2 + 2*a**2*b**4*d**2 + x**4*(32*a**2*c**4*d**2 - 16*a*b**2*c**3*d**2 + 2*b**4*c**2*d**2) + x**3*(
64*a**2*b*c**3*d**2 - 32*a*b**3*c**2*d**2 + 4*b**5*c*d**2) + x**2*(64*a**3*c**3*d**2 - 12*a*b**4*c*d**2 + 2*b*
*6*d**2) + x*(64*a**3*b*c**2*d**2 - 32*a**2*b**3*c*d**2 + 4*a*b**5*d**2))

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Giac [A]
time = 3.47, size = 219, normalized size = 1.56 \begin {gather*} \frac {6 \, {\left (2 \, c^{2} g - b c h\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {12 \, c^{3} g x^{3} - 6 \, b c^{2} h x^{3} + 18 \, b c^{2} g x^{2} - 9 \, b^{2} c h x^{2} + 4 \, b^{2} c g x + 20 \, a c^{2} g x - 2 \, b^{3} h x - 10 \, a b c h x - b^{3} g + 10 \, a b c g - a b^{2} h - 8 \, a^{2} c h}{2 \, {\left (b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2}\right )} {\left (c x^{2} + b x + a\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*(2*c^2*g - b*c*h)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*d^2 - 8*a*b^2*c*d^2 + 16*a^2*c^2*d^2)*sqrt(-b
^2 + 4*a*c)) + 1/2*(12*c^3*g*x^3 - 6*b*c^2*h*x^3 + 18*b*c^2*g*x^2 - 9*b^2*c*h*x^2 + 4*b^2*c*g*x + 20*a*c^2*g*x
- 2*b^3*h*x - 10*a*b*c*h*x - b^3*g + 10*a*b*c*g - a*b^2*h - 8*a^2*c*h)/((b^4*d^2 - 8*a*b^2*c*d^2 + 16*a^2*c^2
*d^2)*(c*x^2 + b*x + a)^2)

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Mupad [B]
time = 0.42, size = 395, normalized size = 2.82 \begin {gather*} \frac {6\,c\,\mathrm {atan}\left (\frac {d^2\,\left (\frac {6\,c^2\,x\,\left (b\,h-2\,c\,g\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{5/2}}+\frac {3\,c\,\left (b\,h-2\,c\,g\right )\,\left (16\,a^2\,b\,c^2\,d^2-8\,a\,b^3\,c\,d^2+b^5\,d^2\right )}{d^4\,{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{6\,c^2\,g-3\,b\,c\,h}\right )\,\left (b\,h-2\,c\,g\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {8\,c\,h\,a^2+h\,a\,b^2-10\,c\,g\,a\,b+g\,b^3}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x\,\left (b^2+5\,a\,c\right )\,\left (b\,h-2\,c\,g\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {3\,c^2\,x^3\,\left (b\,h-2\,c\,g\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {9\,b\,c\,x^2\,\left (b\,h-2\,c\,g\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^2\,\left (b^2\,d^2+2\,a\,c\,d^2\right )+a^2\,d^2+c^2\,d^2\,x^4+2\,a\,b\,d^2\,x+2\,b\,c\,d^2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*c*atan((d^2*((6*c^2*x*(b*h - 2*c*g))/(d^2*(4*a*c - b^2)^(5/2)) + (3*c*(b*h - 2*c*g)*(b^5*d^2 + 16*a^2*b*c^2
*d^2 - 8*a*b^3*c*d^2))/(d^4*(4*a*c - b^2)^(5/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(b^4 + 16*a^2*c^2 - 8*a*b^2*c
))/(6*c^2*g - 3*b*c*h))*(b*h - 2*c*g))/(d^2*(4*a*c - b^2)^(5/2)) - ((b^3*g + a*b^2*h + 8*a^2*c*h - 10*a*b*c*g)
/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x*(5*a*c + b^2)*(b*h - 2*c*g))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (3*c^2*
x^3*(b*h - 2*c*g))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (9*b*c*x^2*(b*h - 2*c*g))/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c
)))/(x^2*(b^2*d^2 + 2*a*c*d^2) + a^2*d^2 + c^2*d^2*x^4 + 2*a*b*d^2*x + 2*b*c*d^2*x^3)

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