\(\int \frac {(b d+2 c d x)^4}{(a+b x+c x^2)^3} \, dx\) [1183]

Optimal result

Integrand size = 24, antiderivative size = 92 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac {3 c d^4 (b+2 c x)}{a+b x+c x^2}-\frac {12 c^2 d^4 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]

[Out]

-1/2*d^4*(2*c*x+b)^3/(c*x^2+b*x+a)^2-3*c*d^4*(2*c*x+b)/(c*x^2+b*x+a)-12*c^2*d^4*arctanh((2*c*x+b)/(-4*a*c+b^2)
^(1/2))/(-4*a*c+b^2)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {700, 632, 212} \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {12 c^2 d^4 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {3 c d^4 (b+2 c x)}{a+b x+c x^2}-\frac {d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2} \]

[In]

Int[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^3,x]

[Out]

-1/2*(d^4*(b + 2*c*x)^3)/(a + b*x + c*x^2)^2 - (3*c*d^4*(b + 2*c*x))/(a + b*x + c*x^2) - (12*c^2*d^4*ArcTanh[(
b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c]

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 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 700

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

Rubi steps \begin{align*} \text {integral}& = -\frac {d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2}+\left (3 c d^2\right ) \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^2} \, dx \\ & = -\frac {d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac {3 c d^4 (b+2 c x)}{a+b x+c x^2}+\left (6 c^2 d^4\right ) \int \frac {1}{a+b x+c x^2} \, dx \\ & = -\frac {d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac {3 c d^4 (b+2 c x)}{a+b x+c x^2}-\left (12 c^2 d^4\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right ) \\ & = -\frac {d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac {3 c d^4 (b+2 c x)}{a+b x+c x^2}-\frac {12 c^2 d^4 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.97 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^3} \, dx=d^4 \left (-\frac {(b+2 c x) \left (b^2+10 b c x+2 c \left (3 a+5 c x^2\right )\right )}{2 (a+x (b+c x))^2}+\frac {12 c^2 \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}\right ) \]

[In]

Integrate[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^3,x]

[Out]

d^4*(-1/2*((b + 2*c*x)*(b^2 + 10*b*c*x + 2*c*(3*a + 5*c*x^2)))/(a + x*(b + c*x))^2 + (12*c^2*ArcTan[(b + 2*c*x
)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])

Maple [A] (verified)

Time = 2.73 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04

method	result	size
default	\(d^{4} \left (\frac {-10 c^{3} x^{3}-15 b \,c^{2} x^{2}-6 c \left (a c +b^{2}\right ) x -\frac {b \left (6 a c +b^{2}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {12 c^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )\)	\(96\)
risch	\(\frac {-10 d^{4} c^{3} x^{3}-15 c^{2} d^{4} b \,x^{2}-6 d^{4} c \left (a c +b^{2}\right ) x -\frac {d^{4} b \left (6 a c +b^{2}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {6 d^{4} c^{2} \ln \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}{\sqrt {-4 a c +b^{2}}}+\frac {6 d^{4} c^{2} \ln \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}{\sqrt {-4 a c +b^{2}}}\)	\(139\)

[In]

int((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

d^4*((-10*c^3*x^3-15*b*c^2*x^2-6*c*(a*c+b^2)*x-1/2*b*(6*a*c+b^2))/(c*x^2+b*x+a)^2+12*c^2/(4*a*c-b^2)^(1/2)*arc
tan((2*c*x+b)/(4*a*c-b^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (86) = 172\).

Time = 0.31 (sec) , antiderivative size = 625, normalized size of antiderivative = 6.79 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^3} \, dx=\left [-\frac {20 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{3} + 30 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x^{2} + 12 \, {\left (b^{4} c - 3 \, a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} x + {\left (b^{5} + 2 \, a b^{3} c - 24 \, a^{2} b c^{2}\right )} d^{4} - 12 \, {\left (c^{4} d^{4} x^{4} + 2 \, b c^{3} d^{4} x^{3} + 2 \, a b c^{2} d^{4} x + a^{2} c^{2} d^{4} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )}}, -\frac {20 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{3} + 30 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x^{2} + 12 \, {\left (b^{4} c - 3 \, a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} x + {\left (b^{5} + 2 \, a b^{3} c - 24 \, a^{2} b c^{2}\right )} d^{4} + 24 \, {\left (c^{4} d^{4} x^{4} + 2 \, b c^{3} d^{4} x^{3} + 2 \, a b c^{2} d^{4} x + a^{2} c^{2} d^{4} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )}}\right ] \]

[In]

integrate((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[-1/2*(20*(b^2*c^3 - 4*a*c^4)*d^4*x^3 + 30*(b^3*c^2 - 4*a*b*c^3)*d^4*x^2 + 12*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3
)*d^4*x + (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^4 - 12*(c^4*d^4*x^4 + 2*b*c^3*d^4*x^3 + 2*a*b*c^2*d^4*x + a^2*c^2
*d^4 + (b^2*c^2 + 2*a*c^3)*d^4*x^2)*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^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 +
(b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x), -1/2*(20*(b^2*c^3 - 4*a*c^4)*d^4*x^3 + 30*(b^3*
c^2 - 4*a*b*c^3)*d^4*x^2 + 12*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*x + (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^4 +
24*(c^4*d^4*x^4 + 2*b*c^3*d^4*x^3 + 2*a*b*c^2*d^4*x + a^2*c^2*d^4 + (b^2*c^2 + 2*a*c^3)*d^4*x^2)*sqrt(-b^2 +
4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)))/((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2
*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (90) = 180\).

Time = 1.15 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.30 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^3} \, dx=- 6 c^{2} d^{4} \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x + \frac {- 24 a c^{3} d^{4} \sqrt {- \frac {1}{4 a c - b^{2}}} + 6 b^{2} c^{2} d^{4} \sqrt {- \frac {1}{4 a c - b^{2}}} + 6 b c^{2} d^{4}}{12 c^{3} d^{4}} \right )} + 6 c^{2} d^{4} \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x + \frac {24 a c^{3} d^{4} \sqrt {- \frac {1}{4 a c - b^{2}}} - 6 b^{2} c^{2} d^{4} \sqrt {- \frac {1}{4 a c - b^{2}}} + 6 b c^{2} d^{4}}{12 c^{3} d^{4}} \right )} + \frac {- 6 a b c d^{4} - b^{3} d^{4} - 30 b c^{2} d^{4} x^{2} - 20 c^{3} d^{4} x^{3} + x \left (- 12 a c^{2} d^{4} - 12 b^{2} c d^{4}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \cdot \left (4 a c + 2 b^{2}\right )} \]

[In]

integrate((2*c*d*x+b*d)**4/(c*x**2+b*x+a)**3,x)

[Out]

-6*c**2*d**4*sqrt(-1/(4*a*c - b**2))*log(x + (-24*a*c**3*d**4*sqrt(-1/(4*a*c - b**2)) + 6*b**2*c**2*d**4*sqrt(
-1/(4*a*c - b**2)) + 6*b*c**2*d**4)/(12*c**3*d**4)) + 6*c**2*d**4*sqrt(-1/(4*a*c - b**2))*log(x + (24*a*c**3*d
**4*sqrt(-1/(4*a*c - b**2)) - 6*b**2*c**2*d**4*sqrt(-1/(4*a*c - b**2)) + 6*b*c**2*d**4)/(12*c**3*d**4)) + (-6*
a*b*c*d**4 - b**3*d**4 - 30*b*c**2*d**4*x**2 - 20*c**3*d**4*x**3 + x*(-12*a*c**2*d**4 - 12*b**2*c*d**4))/(2*a*
*2 + 4*a*b*x + 4*b*c*x**3 + 2*c**2*x**4 + x**2*(4*a*c + 2*b**2))

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^3,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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.24 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^3} \, dx=\frac {12 \, c^{2} d^{4} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {20 \, c^{3} d^{4} x^{3} + 30 \, b c^{2} d^{4} x^{2} + 12 \, b^{2} c d^{4} x + 12 \, a c^{2} d^{4} x + b^{3} d^{4} + 6 \, a b c d^{4}}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} \]

[In]

integrate((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

12*c^2*d^4*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) - 1/2*(20*c^3*d^4*x^3 + 30*b*c^2*d^4*x^2
+ 12*b^2*c*d^4*x + 12*a*c^2*d^4*x + b^3*d^4 + 6*a*b*c*d^4)/(c*x^2 + b*x + a)^2

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.80 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^3} \, dx=\frac {12\,c^2\,d^4\,\mathrm {atan}\left (\frac {\frac {6\,b\,c^2\,d^4}{\sqrt {4\,a\,c-b^2}}+\frac {12\,c^3\,d^4\,x}{\sqrt {4\,a\,c-b^2}}}{6\,c^2\,d^4}\right )}{\sqrt {4\,a\,c-b^2}}-\frac {\frac {b^3\,d^4}{2}+10\,c^3\,d^4\,x^3+15\,b\,c^2\,d^4\,x^2+6\,c\,d^4\,x\,\left (b^2+a\,c\right )+3\,a\,b\,c\,d^4}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \]

[In]

int((b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^3,x)

[Out]

(12*c^2*d^4*atan(((6*b*c^2*d^4)/(4*a*c - b^2)^(1/2) + (12*c^3*d^4*x)/(4*a*c - b^2)^(1/2))/(6*c^2*d^4)))/(4*a*c
- b^2)^(1/2) - ((b^3*d^4)/2 + 10*c^3*d^4*x^3 + 15*b*c^2*d^4*x^2 + 6*c*d^4*x*(a*c + b^2) + 3*a*b*c*d^4)/(x^2*(
2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3)