3.1168 \(\int \frac{(b d+2 c d x)^6}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=100 \[ 20 c d^6 \left (b^2-4 a c\right ) (b+2 c x)-20 c d^6 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^6 (b+2 c x)^5}{a+b x+c x^2}+\frac{20}{3} c d^6 (b+2 c x)^3 \]

[Out]

20*c*(b^2 - 4*a*c)*d^6*(b + 2*c*x) + (20*c*d^6*(b + 2*c*x)^3)/3 - (d^6*(b + 2*c*x)^5)/(a + b*x + c*x^2) - 20*c
*(b^2 - 4*a*c)^(3/2)*d^6*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]

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Rubi [A]  time = 0.0742676, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {686, 692, 618, 206} \[ 20 c d^6 \left (b^2-4 a c\right ) (b+2 c x)-20 c d^6 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^6 (b+2 c x)^5}{a+b x+c x^2}+\frac{20}{3} c d^6 (b+2 c x)^3 \]

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^2,x]

[Out]

20*c*(b^2 - 4*a*c)*d^6*(b + 2*c*x) + (20*c*d^6*(b + 2*c*x)^3)/3 - (d^6*(b + 2*c*x)^5)/(a + b*x + c*x^2) - 20*c
*(b^2 - 4*a*c)^(3/2)*d^6*ArcTanh[(b + 2*c*x)/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 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)^6}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{d^6 (b+2 c x)^5}{a+b x+c x^2}+\left (10 c d^2\right ) \int \frac{(b d+2 c d x)^4}{a+b x+c x^2} \, dx\\ &=\frac{20}{3} c d^6 (b+2 c x)^3-\frac{d^6 (b+2 c x)^5}{a+b x+c x^2}+\left (10 c \left (b^2-4 a c\right ) d^4\right ) \int \frac{(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=20 c \left (b^2-4 a c\right ) d^6 (b+2 c x)+\frac{20}{3} c d^6 (b+2 c x)^3-\frac{d^6 (b+2 c x)^5}{a+b x+c x^2}+\left (10 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac{1}{a+b x+c x^2} \, dx\\ &=20 c \left (b^2-4 a c\right ) d^6 (b+2 c x)+\frac{20}{3} c d^6 (b+2 c x)^3-\frac{d^6 (b+2 c x)^5}{a+b x+c x^2}-\left (20 c \left (b^2-4 a c\right )^2 d^6\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=20 c \left (b^2-4 a c\right ) d^6 (b+2 c x)+\frac{20}{3} c d^6 (b+2 c x)^3-\frac{d^6 (b+2 c x)^5}{a+b x+c x^2}-20 c \left (b^2-4 a c\right )^{3/2} d^6 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0654078, size = 108, normalized size = 1.08 \[ d^6 \left (-16 c^2 x \left (8 a c-3 b^2\right )-\frac{\left (b^2-4 a c\right )^2 (b+2 c x)}{a+x (b+c x)}+20 c \left (4 a c-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+32 b c^3 x^2+\frac{64 c^4 x^3}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^2,x]

[Out]

d^6*(-16*c^2*(-3*b^2 + 8*a*c)*x + 32*b*c^3*x^2 + (64*c^4*x^3)/3 - ((b^2 - 4*a*c)^2*(b + 2*c*x))/(a + x*(b + c*
x)) + 20*c*(-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 = 0.157, size = 312, normalized size = 3.1 \begin{align*}{\frac{64\,{d}^{6}{c}^{4}{x}^{3}}{3}}+32\,{d}^{6}b{c}^{3}{x}^{2}-128\,{d}^{6}a{c}^{3}x+48\,{d}^{6}{b}^{2}{c}^{2}x-32\,{\frac{{d}^{6}{a}^{2}{c}^{3}x}{c{x}^{2}+bx+a}}+16\,{\frac{{d}^{6}a{b}^{2}{c}^{2}x}{c{x}^{2}+bx+a}}-2\,{\frac{{d}^{6}c{b}^{4}x}{c{x}^{2}+bx+a}}-16\,{\frac{{d}^{6}{a}^{2}b{c}^{2}}{c{x}^{2}+bx+a}}+8\,{\frac{{d}^{6}a{b}^{3}c}{c{x}^{2}+bx+a}}-{\frac{{d}^{6}{b}^{5}}{c{x}^{2}+bx+a}}+320\,{\frac{{d}^{6}{a}^{2}{c}^{3}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-160\,{\frac{{d}^{6}a{b}^{2}{c}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+20\,{\frac{{d}^{6}c{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)^6/(c*x^2+b*x+a)^2,x)

[Out]

64/3*d^6*c^4*x^3+32*d^6*b*c^3*x^2-128*d^6*a*c^3*x+48*d^6*b^2*c^2*x-32*d^6/(c*x^2+b*x+a)*a^2*c^3*x+16*d^6/(c*x^
2+b*x+a)*a*b^2*c^2*x-2*d^6/(c*x^2+b*x+a)*c*b^4*x-16*d^6/(c*x^2+b*x+a)*a^2*b*c^2+8*d^6/(c*x^2+b*x+a)*a*b^3*c-d^
6/(c*x^2+b*x+a)*b^5+320*d^6*c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2-160*d^6*c^2/(4*a*c-b
^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2+20*d^6*c/(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)^6/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.02972, size = 1085, normalized size = 10.85 \begin{align*} \left [\frac{64 \, c^{5} d^{6} x^{5} + 160 \, b c^{4} d^{6} x^{4} + 80 \,{\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 144 \,{\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} d^{6} x^{2} - 6 \,{\left (b^{4} c - 32 \, a b^{2} c^{2} + 80 \, a^{2} c^{3}\right )} d^{6} x - 3 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{6} - 30 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{6} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{6} x +{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{6}\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 )}{3 \,{\left (c x^{2} + b x + a\right )}}, \frac{64 \, c^{5} d^{6} x^{5} + 160 \, b c^{4} d^{6} x^{4} + 80 \,{\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 144 \,{\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} d^{6} x^{2} - 6 \,{\left (b^{4} c - 32 \, a b^{2} c^{2} + 80 \, a^{2} c^{3}\right )} d^{6} x - 3 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{6} - 60 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{6} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{6} x +{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{6}\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 )}{3 \,{\left (c x^{2} + b x + a\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

[1/3*(64*c^5*d^6*x^5 + 160*b*c^4*d^6*x^4 + 80*(3*b^2*c^3 - 4*a*c^4)*d^6*x^3 + 144*(b^3*c^2 - 2*a*b*c^3)*d^6*x^
2 - 6*(b^4*c - 32*a*b^2*c^2 + 80*a^2*c^3)*d^6*x - 3*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^6 - 30*((b^2*c^2 - 4*a*
c^3)*d^6*x^2 + (b^3*c - 4*a*b*c^2)*d^6*x + (a*b^2*c - 4*a^2*c^2)*d^6)*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)))/(c*x^2 + b*x + a), 1/3*(64*c^5*d^6*x^5 +
160*b*c^4*d^6*x^4 + 80*(3*b^2*c^3 - 4*a*c^4)*d^6*x^3 + 144*(b^3*c^2 - 2*a*b*c^3)*d^6*x^2 - 6*(b^4*c - 32*a*b^
2*c^2 + 80*a^2*c^3)*d^6*x - 3*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^6 - 60*((b^2*c^2 - 4*a*c^3)*d^6*x^2 + (b^3*c
- 4*a*b*c^2)*d^6*x + (a*b^2*c - 4*a^2*c^2)*d^6)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2
- 4*a*c)))/(c*x^2 + b*x + a)]

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Sympy [B]  time = 2.20969, size = 313, normalized size = 3.13 \begin{align*} 32 b c^{3} d^{6} x^{2} + \frac{64 c^{4} d^{6} x^{3}}{3} - 10 c d^{6} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{40 a b c^{2} d^{6} - 10 b^{3} c d^{6} - 10 c d^{6} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{80 a c^{3} d^{6} - 20 b^{2} c^{2} d^{6}} \right )} + 10 c d^{6} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{40 a b c^{2} d^{6} - 10 b^{3} c d^{6} + 10 c d^{6} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{80 a c^{3} d^{6} - 20 b^{2} c^{2} d^{6}} \right )} + x \left (- 128 a c^{3} d^{6} + 48 b^{2} c^{2} d^{6}\right ) - \frac{16 a^{2} b c^{2} d^{6} - 8 a b^{3} c d^{6} + b^{5} d^{6} + x \left (32 a^{2} c^{3} d^{6} - 16 a b^{2} c^{2} d^{6} + 2 b^{4} c d^{6}\right )}{a + b x + c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**6/(c*x**2+b*x+a)**2,x)

[Out]

32*b*c**3*d**6*x**2 + 64*c**4*d**6*x**3/3 - 10*c*d**6*sqrt(-(4*a*c - b**2)**3)*log(x + (40*a*b*c**2*d**6 - 10*
b**3*c*d**6 - 10*c*d**6*sqrt(-(4*a*c - b**2)**3))/(80*a*c**3*d**6 - 20*b**2*c**2*d**6)) + 10*c*d**6*sqrt(-(4*a
*c - b**2)**3)*log(x + (40*a*b*c**2*d**6 - 10*b**3*c*d**6 + 10*c*d**6*sqrt(-(4*a*c - b**2)**3))/(80*a*c**3*d**
6 - 20*b**2*c**2*d**6)) + x*(-128*a*c**3*d**6 + 48*b**2*c**2*d**6) - (16*a**2*b*c**2*d**6 - 8*a*b**3*c*d**6 +
b**5*d**6 + x*(32*a**2*c**3*d**6 - 16*a*b**2*c**2*d**6 + 2*b**4*c*d**6))/(a + b*x + c*x**2)

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Giac [B]  time = 1.18774, size = 266, normalized size = 2.66 \begin{align*} \frac{20 \,{\left (b^{4} c d^{6} - 8 \, a b^{2} c^{2} d^{6} + 16 \, a^{2} c^{3} d^{6}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, b^{4} c d^{6} x - 16 \, a b^{2} c^{2} d^{6} x + 32 \, a^{2} c^{3} d^{6} x + b^{5} d^{6} - 8 \, a b^{3} c d^{6} + 16 \, a^{2} b c^{2} d^{6}}{c x^{2} + b x + a} + \frac{16 \,{\left (4 \, c^{10} d^{6} x^{3} + 6 \, b c^{9} d^{6} x^{2} + 9 \, b^{2} c^{8} d^{6} x - 24 \, a c^{9} d^{6} x\right )}}{3 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

20*(b^4*c*d^6 - 8*a*b^2*c^2*d^6 + 16*a^2*c^3*d^6)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) -
(2*b^4*c*d^6*x - 16*a*b^2*c^2*d^6*x + 32*a^2*c^3*d^6*x + b^5*d^6 - 8*a*b^3*c*d^6 + 16*a^2*b*c^2*d^6)/(c*x^2 +
b*x + a) + 16/3*(4*c^10*d^6*x^3 + 6*b*c^9*d^6*x^2 + 9*b^2*c^8*d^6*x - 24*a*c^9*d^6*x)/c^6