3.1181 \(\int \frac{(b d+2 c d x)^6}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=108 \[ -60 c^2 d^6 \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{5 c d^6 (b+2 c x)^3}{a+b x+c x^2}-\frac{d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}+60 c^2 d^6 (b+2 c x) \]

[Out]

60*c^2*d^6*(b + 2*c*x) - (d^6*(b + 2*c*x)^5)/(2*(a + b*x + c*x^2)^2) - (5*c*d^6*(b + 2*c*x)^3)/(a + b*x + c*x^
2) - 60*c^2*Sqrt[b^2 - 4*a*c]*d^6*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]

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Rubi [A]  time = 0.0726404, antiderivative size = 108, 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} \[ -60 c^2 d^6 \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{5 c d^6 (b+2 c x)^3}{a+b x+c x^2}-\frac{d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}+60 c^2 d^6 (b+2 c x) \]

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^3,x]

[Out]

60*c^2*d^6*(b + 2*c*x) - (d^6*(b + 2*c*x)^5)/(2*(a + b*x + c*x^2)^2) - (5*c*d^6*(b + 2*c*x)^3)/(a + b*x + c*x^
2) - 60*c^2*Sqrt[b^2 - 4*a*c]*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 )^3} \, dx &=-\frac{d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}+\left (5 c d^2\right ) \int \frac{(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac{5 c d^6 (b+2 c x)^3}{a+b x+c x^2}+\left (30 c^2 d^4\right ) \int \frac{(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=60 c^2 d^6 (b+2 c x)-\frac{d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac{5 c d^6 (b+2 c x)^3}{a+b x+c x^2}+\left (30 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac{1}{a+b x+c x^2} \, dx\\ &=60 c^2 d^6 (b+2 c x)-\frac{d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac{5 c d^6 (b+2 c x)^3}{a+b x+c x^2}-\left (60 c^2 \left (b^2-4 a c\right ) d^6\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=60 c^2 d^6 (b+2 c x)-\frac{d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac{5 c d^6 (b+2 c x)^3}{a+b x+c x^2}-60 c^2 \sqrt{b^2-4 a c} d^6 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0650986, size = 113, normalized size = 1.05 \[ d^6 \left (-60 c^2 \sqrt{4 a c-b^2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+\frac{9 c \left (4 a c-b^2\right ) (b+2 c x)}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^2 (b+2 c x)}{2 (a+x (b+c x))^2}+64 c^3 x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^3,x]

[Out]

d^6*(64*c^3*x - ((b^2 - 4*a*c)^2*(b + 2*c*x))/(2*(a + x*(b + c*x))^2) + (9*c*(-b^2 + 4*a*c)*(b + 2*c*x))/(a +
x*(b + c*x)) - 60*c^2*Sqrt[-b^2 + 4*a*c]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])

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Maple [B]  time = 0.157, size = 289, normalized size = 2.7 \begin{align*} 64\,{d}^{6}{c}^{3}x+72\,{\frac{{d}^{6}{x}^{3}a{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-18\,{\frac{{d}^{6}{x}^{3}{b}^{2}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+108\,{\frac{{d}^{6}{x}^{2}ab{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-27\,{\frac{{d}^{6}{x}^{2}{b}^{3}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+56\,{\frac{{d}^{6}{a}^{2}{c}^{3}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+26\,{\frac{{d}^{6}a{b}^{2}{c}^{2}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-10\,{\frac{{d}^{6}c{b}^{4}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+28\,{\frac{{d}^{6}{a}^{2}b{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-5\,{\frac{{d}^{6}a{b}^{3}c}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{6}{b}^{5}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}-60\,{d}^{6}{c}^{2}\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)^3,x)

[Out]

64*d^6*c^3*x+72*d^6/(c*x^2+b*x+a)^2*x^3*a*c^4-18*d^6/(c*x^2+b*x+a)^2*x^3*b^2*c^3+108*d^6/(c*x^2+b*x+a)^2*x^2*a
*b*c^3-27*d^6/(c*x^2+b*x+a)^2*x^2*b^3*c^2+56*d^6/(c*x^2+b*x+a)^2*a^2*c^3*x+26*d^6/(c*x^2+b*x+a)^2*a*b^2*c^2*x-
10*d^6/(c*x^2+b*x+a)^2*c*b^4*x+28*d^6/(c*x^2+b*x+a)^2*a^2*b*c^2-5*d^6/(c*x^2+b*x+a)^2*a*b^3*c-1/2*d^6/(c*x^2+b
*x+a)^2*b^5-60*d^6*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)^6/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.3224, size = 1235, normalized size = 11.44 \begin{align*} \left [\frac{128 \, c^{5} d^{6} x^{5} + 256 \, b c^{4} d^{6} x^{4} + 4 \,{\left (23 \, b^{2} c^{3} + 100 \, a c^{4}\right )} d^{6} x^{3} - 2 \,{\left (27 \, b^{3} c^{2} - 236 \, a b c^{3}\right )} d^{6} x^{2} - 4 \,{\left (5 \, b^{4} c - 13 \, a b^{2} c^{2} - 60 \, a^{2} c^{3}\right )} d^{6} x -{\left (b^{5} + 10 \, a b^{3} c - 56 \, a^{2} b c^{2}\right )} d^{6} + 60 \,{\left (c^{4} d^{6} x^{4} + 2 \, b c^{3} d^{6} x^{3} + 2 \, a b c^{2} d^{6} x + a^{2} c^{2} d^{6} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} 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 (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, \frac{128 \, c^{5} d^{6} x^{5} + 256 \, b c^{4} d^{6} x^{4} + 4 \,{\left (23 \, b^{2} c^{3} + 100 \, a c^{4}\right )} d^{6} x^{3} - 2 \,{\left (27 \, b^{3} c^{2} - 236 \, a b c^{3}\right )} d^{6} x^{2} - 4 \,{\left (5 \, b^{4} c - 13 \, a b^{2} c^{2} - 60 \, a^{2} c^{3}\right )} d^{6} x -{\left (b^{5} + 10 \, a b^{3} c - 56 \, a^{2} b c^{2}\right )} d^{6} - 120 \,{\left (c^{4} d^{6} x^{4} + 2 \, b c^{3} d^{6} x^{3} + 2 \, a b c^{2} d^{6} x + a^{2} c^{2} d^{6} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} 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 (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\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)^3,x, algorithm="fricas")

[Out]

[1/2*(128*c^5*d^6*x^5 + 256*b*c^4*d^6*x^4 + 4*(23*b^2*c^3 + 100*a*c^4)*d^6*x^3 - 2*(27*b^3*c^2 - 236*a*b*c^3)*
d^6*x^2 - 4*(5*b^4*c - 13*a*b^2*c^2 - 60*a^2*c^3)*d^6*x - (b^5 + 10*a*b^3*c - 56*a^2*b*c^2)*d^6 + 60*(c^4*d^6*
x^4 + 2*b*c^3*d^6*x^3 + 2*a*b*c^2*d^6*x + a^2*c^2*d^6 + (b^2*c^2 + 2*a*c^3)*d^6*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)))/(c^2*x^4 + 2*b*c*x^3 + 2*
a*b*x + (b^2 + 2*a*c)*x^2 + a^2), 1/2*(128*c^5*d^6*x^5 + 256*b*c^4*d^6*x^4 + 4*(23*b^2*c^3 + 100*a*c^4)*d^6*x^
3 - 2*(27*b^3*c^2 - 236*a*b*c^3)*d^6*x^2 - 4*(5*b^4*c - 13*a*b^2*c^2 - 60*a^2*c^3)*d^6*x - (b^5 + 10*a*b^3*c -
56*a^2*b*c^2)*d^6 - 120*(c^4*d^6*x^4 + 2*b*c^3*d^6*x^3 + 2*a*b*c^2*d^6*x + a^2*c^2*d^6 + (b^2*c^2 + 2*a*c^3)*
d^6*x^2)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)))/(c^2*x^4 + 2*b*c*x^3 + 2*a*
b*x + (b^2 + 2*a*c)*x^2 + a^2)]

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Sympy [B]  time = 7.11762, size = 299, normalized size = 2.77 \begin{align*} 64 c^{3} d^{6} x + c^{2} d^{6} \sqrt{- 3600 a c + 900 b^{2}} \log{\left (x + \frac{30 b c^{2} d^{6} - c^{2} d^{6} \sqrt{- 3600 a c + 900 b^{2}}}{60 c^{3} d^{6}} \right )} - c^{2} d^{6} \sqrt{- 3600 a c + 900 b^{2}} \log{\left (x + \frac{30 b c^{2} d^{6} + c^{2} d^{6} \sqrt{- 3600 a c + 900 b^{2}}}{60 c^{3} d^{6}} \right )} + \frac{56 a^{2} b c^{2} d^{6} - 10 a b^{3} c d^{6} - b^{5} d^{6} + x^{3} \left (144 a c^{4} d^{6} - 36 b^{2} c^{3} d^{6}\right ) + x^{2} \left (216 a b c^{3} d^{6} - 54 b^{3} c^{2} d^{6}\right ) + x \left (112 a^{2} c^{3} d^{6} + 52 a b^{2} c^{2} d^{6} - 20 b^{4} c d^{6}\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)**6/(c*x**2+b*x+a)**3,x)

[Out]

64*c**3*d**6*x + c**2*d**6*sqrt(-3600*a*c + 900*b**2)*log(x + (30*b*c**2*d**6 - c**2*d**6*sqrt(-3600*a*c + 900
*b**2))/(60*c**3*d**6)) - c**2*d**6*sqrt(-3600*a*c + 900*b**2)*log(x + (30*b*c**2*d**6 + c**2*d**6*sqrt(-3600*
a*c + 900*b**2))/(60*c**3*d**6)) + (56*a**2*b*c**2*d**6 - 10*a*b**3*c*d**6 - b**5*d**6 + x**3*(144*a*c**4*d**6
- 36*b**2*c**3*d**6) + x**2*(216*a*b*c**3*d**6 - 54*b**3*c**2*d**6) + x*(112*a**2*c**3*d**6 + 52*a*b**2*c**2*
d**6 - 20*b**4*c*d**6))/(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 = 1.17917, size = 265, normalized size = 2.45 \begin{align*} 64 \, c^{3} d^{6} x + \frac{60 \,{\left (b^{2} c^{2} d^{6} - 4 \, a 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{36 \, b^{2} c^{3} d^{6} x^{3} - 144 \, a c^{4} d^{6} x^{3} + 54 \, b^{3} c^{2} d^{6} x^{2} - 216 \, a b c^{3} d^{6} x^{2} + 20 \, b^{4} c d^{6} x - 52 \, a b^{2} c^{2} d^{6} x - 112 \, a^{2} c^{3} d^{6} x + b^{5} d^{6} + 10 \, a b^{3} c d^{6} - 56 \, a^{2} b c^{2} d^{6}}{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)^6/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

64*c^3*d^6*x + 60*(b^2*c^2*d^6 - 4*a*c^3*d^6)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) - 1/2*
(36*b^2*c^3*d^6*x^3 - 144*a*c^4*d^6*x^3 + 54*b^3*c^2*d^6*x^2 - 216*a*b*c^3*d^6*x^2 + 20*b^4*c*d^6*x - 52*a*b^2
*c^2*d^6*x - 112*a^2*c^3*d^6*x + b^5*d^6 + 10*a*b^3*c*d^6 - 56*a^2*b*c^2*d^6)/(c*x^2 + b*x + a)^2