### 3.1183 $$\int \frac{(b d+2 c d x)^4}{(a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=92 $-\frac{12 c^2 d^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{3 c d^4 (b+2 c x)}{a+b x+c x^2}-\frac{d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2}$

[Out]

-(d^4*(b + 2*c*x)^3)/(2*(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]

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Rubi [A]  time = 0.0566576, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {686, 618, 206} $-\frac{12 c^2 d^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{3 c d^4 (b+2 c x)}{a+b x+c x^2}-\frac{d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(d^4*(b + 2*c*x)^3)/(2*(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 686

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]

Rule 618

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 206

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

Rubi steps

\begin{align*} \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}+\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 ) \operatorname{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]  time = 0.0647679, size = 89, normalized size = 0.97 $d^4 \left (\frac{12 c^2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{(b+2 c x) \left (2 c \left (3 a+5 c x^2\right )+b^2+10 b c x\right )}{2 (a+x (b+c x))^2}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

d^4*(-((b + 2*c*x)*(b^2 + 10*b*c*x + 2*c*(3*a + 5*c*x^2)))/(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])

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Maple [A]  time = 0.157, size = 173, normalized size = 1.9 \begin{align*} -10\,{\frac{{d}^{4}{c}^{3}{x}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-15\,{\frac{{d}^{4}b{c}^{2}{x}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-6\,{\frac{{d}^{4}a{c}^{2}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-6\,{\frac{{d}^{4}{b}^{2}cx}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-3\,{\frac{{d}^{4}abc}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{4}{b}^{3}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}+12\,{\frac{{c}^{2}{d}^{4}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.26966, size = 1312, normalized size = 14.26 \begin{align*} \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 ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [B]  time = 3.96293, size = 303, normalized size = 3.29 \begin{align*} - 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} \left (4 a c + 2 b^{2}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.26437, size = 154, normalized size = 1.67 \begin{align*} \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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