### 3.1158 $$\int \frac{(b d+2 c d x)^4}{a+b x+c x^2} \, dx$$

Optimal. Leaf size=72 $2 d^4 \left (b^2-4 a c\right ) (b+2 c x)-2 d^4 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+\frac{2}{3} d^4 (b+2 c x)^3$

[Out]

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

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Rubi [A]  time = 0.070038, antiderivative size = 72, 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 = {692, 618, 206} $2 d^4 \left (b^2-4 a c\right ) (b+2 c x)-2 d^4 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+\frac{2}{3} d^4 (b+2 c x)^3$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 692

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

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

Mathematica [A]  time = 0.0418401, size = 72, normalized size = 1. $d^4 \left (\frac{8}{3} c x \left (2 c \left (c x^2-3 a\right )+3 b^2+3 b c x\right )+2 \left (4 a c-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

d^4*((8*c*x*(3*b^2 + 3*b*c*x + 2*c*(-3*a + c*x^2)))/3 + 2*(-b^2 + 4*a*c)^(3/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 +
4*a*c]])

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Maple [B]  time = 0.147, size = 170, normalized size = 2.4 \begin{align*}{\frac{16\,{d}^{4}{c}^{3}{x}^{3}}{3}}+8\,{d}^{4}b{c}^{2}{x}^{2}-16\,{d}^{4}a{c}^{2}x+8\,{d}^{4}{b}^{2}cx+32\,{\frac{{d}^{4}{a}^{2}{c}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-16\,{\frac{{d}^{4}ac{b}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{d}^{4}{b}^{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),x)

[Out]

16/3*d^4*c^3*x^3+8*d^4*b*c^2*x^2-16*d^4*a*c^2*x+8*d^4*b^2*c*x+32*d^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c
-b^2)^(1/2))*a^2*c^2-16*d^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c*b^2+2*d^4/(4*a*c-b^2)^(1
/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72199, size = 464, normalized size = 6.44 \begin{align*} \left [\frac{16}{3} \, c^{3} d^{4} x^{3} + 8 \, b c^{2} d^{4} x^{2} -{\left (b^{2} - 4 \, a c\right )}^{\frac{3}{2}} d^{4} \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 ) + 8 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d^{4} x, \frac{16}{3} \, c^{3} d^{4} x^{3} + 8 \, b c^{2} d^{4} x^{2} - 2 \,{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c} d^{4} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 8 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d^{4} x\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),x, algorithm="fricas")

[Out]

[16/3*c^3*d^4*x^3 + 8*b*c^2*d^4*x^2 - (b^2 - 4*a*c)^(3/2)*d^4*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)) + 8*(b^2*c - 2*a*c^2)*d^4*x, 16/3*c^3*d^4*x^3 + 8*b*c^2*d^4*x^2 - 2
*(b^2 - 4*a*c)*sqrt(-b^2 + 4*a*c)*d^4*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 8*(b^2*c - 2*a*c
^2)*d^4*x]

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Sympy [B]  time = 0.745157, size = 204, normalized size = 2.83 \begin{align*} 8 b c^{2} d^{4} x^{2} + \frac{16 c^{3} d^{4} x^{3}}{3} - d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{4 a b c d^{4} - b^{3} d^{4} - d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{8 a c^{2} d^{4} - 2 b^{2} c d^{4}} \right )} + d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{4 a b c d^{4} - b^{3} d^{4} + d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{8 a c^{2} d^{4} - 2 b^{2} c d^{4}} \right )} + x \left (- 16 a c^{2} d^{4} + 8 b^{2} c d^{4}\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),x)

[Out]

8*b*c**2*d**4*x**2 + 16*c**3*d**4*x**3/3 - d**4*sqrt(-(4*a*c - b**2)**3)*log(x + (4*a*b*c*d**4 - b**3*d**4 - d
**4*sqrt(-(4*a*c - b**2)**3))/(8*a*c**2*d**4 - 2*b**2*c*d**4)) + d**4*sqrt(-(4*a*c - b**2)**3)*log(x + (4*a*b*
c*d**4 - b**3*d**4 + d**4*sqrt(-(4*a*c - b**2)**3))/(8*a*c**2*d**4 - 2*b**2*c*d**4)) + x*(-16*a*c**2*d**4 + 8*
b**2*c*d**4)

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Giac [A]  time = 1.11598, size = 155, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} + \frac{8 \,{\left (2 \, c^{6} d^{4} x^{3} + 3 \, b c^{5} d^{4} x^{2} + 3 \, b^{2} c^{4} d^{4} x - 6 \, a c^{5} d^{4} x\right )}}{3 \, c^{3}} \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),x, algorithm="giac")

[Out]

2*(b^4*d^4 - 8*a*b^2*c*d^4 + 16*a^2*c^2*d^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) + 8/3*(
2*c^6*d^4*x^3 + 3*b*c^5*d^4*x^2 + 3*b^2*c^4*d^4*x - 6*a*c^5*d^4*x)/c^3