### 3.1177 $$\int \frac{(b d+2 c d x)^{10}}{(a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=160 $84 c^2 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^3+252 c^2 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)-252 c^2 d^{10} \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\frac{252}{5} c^2 d^{10} (b+2 c x)^5$

[Out]

252*c^2*(b^2 - 4*a*c)^2*d^10*(b + 2*c*x) + 84*c^2*(b^2 - 4*a*c)*d^10*(b + 2*c*x)^3 + (252*c^2*d^10*(b + 2*c*x)
^5)/5 - (d^10*(b + 2*c*x)^9)/(2*(a + b*x + c*x^2)^2) - (9*c*d^10*(b + 2*c*x)^7)/(a + b*x + c*x^2) - 252*c^2*(b
^2 - 4*a*c)^(5/2)*d^10*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]

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Rubi [A]  time = 0.13716, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {686, 692, 618, 206} $84 c^2 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^3+252 c^2 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)-252 c^2 d^{10} \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\frac{252}{5} c^2 d^{10} (b+2 c x)^5$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^10/(a + b*x + c*x^2)^3,x]

[Out]

252*c^2*(b^2 - 4*a*c)^2*d^10*(b + 2*c*x) + 84*c^2*(b^2 - 4*a*c)*d^10*(b + 2*c*x)^3 + (252*c^2*d^10*(b + 2*c*x)
^5)/5 - (d^10*(b + 2*c*x)^9)/(2*(a + b*x + c*x^2)^2) - (9*c*d^10*(b + 2*c*x)^7)/(a + b*x + c*x^2) - 252*c^2*(b
^2 - 4*a*c)^(5/2)*d^10*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)^{10}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\left (9 c d^2\right ) \int \frac{(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 d^4\right ) \int \frac{(b d+2 c d x)^6}{a+b x+c x^2} \, dx\\ &=\frac{252}{5} c^2 d^{10} (b+2 c x)^5-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac{(b d+2 c d x)^4}{a+b x+c x^2} \, dx\\ &=84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac{252}{5} c^2 d^{10} (b+2 c x)^5-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac{(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=252 c^2 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)+84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac{252}{5} c^2 d^{10} (b+2 c x)^5-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 \left (b^2-4 a c\right )^3 d^{10}\right ) \int \frac{1}{a+b x+c x^2} \, dx\\ &=252 c^2 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)+84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac{252}{5} c^2 d^{10} (b+2 c x)^5-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\left (252 c^2 \left (b^2-4 a c\right )^3 d^{10}\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=252 c^2 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)+84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac{252}{5} c^2 d^{10} (b+2 c x)^5-\frac{d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-252 c^2 \left (b^2-4 a c\right )^{5/2} d^{10} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )\\ \end{align*}

Mathematica [A]  time = 0.115103, size = 192, normalized size = 1.2 $d^{10} \left (128 c^3 x \left (48 a^2 c^2-30 a b^2 c+5 b^4\right )-256 c^5 x^3 \left (4 a c-3 b^2\right )+128 b c^4 x^2 \left (5 b^2-12 a c\right )-252 c^2 \left (4 a c-b^2\right )^{5/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+\frac{17 c \left (4 a c-b^2\right )^3 (b+2 c x)}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^4 (b+2 c x)}{2 (a+x (b+c x))^2}+512 b c^6 x^4+\frac{1024 c^7 x^5}{5}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^10/(a + b*x + c*x^2)^3,x]

[Out]

d^10*(128*c^3*(5*b^4 - 30*a*b^2*c + 48*a^2*c^2)*x + 128*b*c^4*(5*b^2 - 12*a*c)*x^2 - 256*c^5*(-3*b^2 + 4*a*c)*
x^3 + 512*b*c^6*x^4 + (1024*c^7*x^5)/5 - ((b^2 - 4*a*c)^4*(b + 2*c*x))/(2*(a + x*(b + c*x))^2) + (17*c*(-b^2 +
4*a*c)^3*(b + 2*c*x))/(a + x*(b + c*x)) - 252*c^2*(-b^2 + 4*a*c)^(5/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]
)

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Maple [B]  time = 0.163, size = 751, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^10/(c*x^2+b*x+a)^3,x)

[Out]

6144*d^10*a^2*c^5*x+640*d^10*b^4*c^3*x+512*d^10*b*c^6*x^4-1024*d^10*x^3*a*c^6+768*d^10*x^3*b^2*c^5+640*d^10*x^
2*b^3*c^4+408*d^10/(c*x^2+b*x+a)^2*x^3*a*b^4*c^4+3264*d^10/(c*x^2+b*x+a)^2*x^2*a^3*b*c^5-1632*d^10/(c*x^2+b*x+
a)^2*x^3*a^2*b^2*c^5-1536*d^10*x^2*a*b*c^5-3840*d^10*b^2*a*c^4*x-16128*d^10*c^5/(4*a*c-b^2)^(1/2)*arctan((2*c*
x+b)/(4*a*c-b^2)^(1/2))*a^3+252*d^10*c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^6+2176*d^10/(
c*x^2+b*x+a)^2*x^3*a^3*c^6-34*d^10/(c*x^2+b*x+a)^2*x^3*b^6*c^3-51*d^10/(c*x^2+b*x+a)^2*x^2*b^7*c^2+1920*d^10/(
c*x^2+b*x+a)^2*x*a^4*c^5-18*d^10/(c*x^2+b*x+a)^2*x*b^8*c+960*d^10/(c*x^2+b*x+a)^2*a^4*b*c^4-688*d^10/(c*x^2+b*
x+a)^2*a^3*b^3*c^3+156*d^10/(c*x^2+b*x+a)^2*a^2*b^5*c^2-9*d^10/(c*x^2+b*x+a)^2*a*b^7*c-3024*d^10*c^3/(4*a*c-b^
2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^4-2448*d^10/(c*x^2+b*x+a)^2*x^2*a^2*b^3*c^4+612*d^10/(c*x^2+b
*x+a)^2*x^2*a*b^5*c^3-288*d^10/(c*x^2+b*x+a)^2*x*a^3*b^2*c^4-504*d^10/(c*x^2+b*x+a)^2*x*a^2*b^4*c^3+186*d^10/(
c*x^2+b*x+a)^2*x*a*b^6*c^2+12096*d^10*c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b^2+1024/5
*d^10*c^7*x^5-1/2*d^10/(c*x^2+b*x+a)^2*b^9

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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)^10/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.16704, size = 2643, normalized size = 16.52 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^10/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[1/10*(2048*c^9*d^10*x^9 + 9216*b*c^8*d^10*x^8 + 1536*(13*b^2*c^7 - 4*a*c^8)*d^10*x^7 + 5376*(5*b^3*c^6 - 4*a*
b*c^7)*d^10*x^6 + 5376*(5*b^4*c^5 - 10*a*b^2*c^6 + 8*a^2*c^7)*d^10*x^5 + 6400*(3*b^5*c^4 - 10*a*b^3*c^5 + 12*a
^2*b*c^6)*d^10*x^4 + 20*(303*b^6*c^3 - 436*a*b^4*c^4 - 2736*a^2*b^2*c^5 + 6720*a^3*c^6)*d^10*x^3 - 10*(51*b^7*
c^2 - 1892*a*b^5*c^3 + 9488*a^2*b^3*c^4 - 14016*a^3*b*c^5)*d^10*x^2 - 20*(9*b^8*c - 93*a*b^6*c^2 - 68*a^2*b^4*
c^3 + 2064*a^3*b^2*c^4 - 4032*a^4*c^5)*d^10*x - 5*(b^9 + 18*a*b^7*c - 312*a^2*b^5*c^2 + 1376*a^3*b^3*c^3 - 192
0*a^4*b*c^4)*d^10 + 1260*((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^10*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*
c^5)*d^10*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^10*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d
^10*x + (a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^10)*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), 1/10*(2048*c^9*d^10*x^9 + 9216*b*c^8*d^10*x^8 + 1536*(13*b^2*c^7 - 4*a*c^8)*d^10*x^7 + 5376*(5*b^3*c^6 -
4*a*b*c^7)*d^10*x^6 + 5376*(5*b^4*c^5 - 10*a*b^2*c^6 + 8*a^2*c^7)*d^10*x^5 + 6400*(3*b^5*c^4 - 10*a*b^3*c^5 +
12*a^2*b*c^6)*d^10*x^4 + 20*(303*b^6*c^3 - 436*a*b^4*c^4 - 2736*a^2*b^2*c^5 + 6720*a^3*c^6)*d^10*x^3 - 10*(51
*b^7*c^2 - 1892*a*b^5*c^3 + 9488*a^2*b^3*c^4 - 14016*a^3*b*c^5)*d^10*x^2 - 20*(9*b^8*c - 93*a*b^6*c^2 - 68*a^2
*b^4*c^3 + 2064*a^3*b^2*c^4 - 4032*a^4*c^5)*d^10*x - 5*(b^9 + 18*a*b^7*c - 312*a^2*b^5*c^2 + 1376*a^3*b^3*c^3
- 1920*a^4*b*c^4)*d^10 - 2520*((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^10*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a
^2*b*c^5)*d^10*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^10*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c
^4)*d^10*x + (a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^10)*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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**10/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.1988, size = 621, normalized size = 3.88 \begin{align*} \frac{252 \,{\left (b^{6} c^{2} d^{10} - 12 \, a b^{4} c^{3} d^{10} + 48 \, a^{2} b^{2} c^{4} d^{10} - 64 \, a^{3} c^{5} d^{10}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{68 \, b^{6} c^{3} d^{10} x^{3} - 816 \, a b^{4} c^{4} d^{10} x^{3} + 3264 \, a^{2} b^{2} c^{5} d^{10} x^{3} - 4352 \, a^{3} c^{6} d^{10} x^{3} + 102 \, b^{7} c^{2} d^{10} x^{2} - 1224 \, a b^{5} c^{3} d^{10} x^{2} + 4896 \, a^{2} b^{3} c^{4} d^{10} x^{2} - 6528 \, a^{3} b c^{5} d^{10} x^{2} + 36 \, b^{8} c d^{10} x - 372 \, a b^{6} c^{2} d^{10} x + 1008 \, a^{2} b^{4} c^{3} d^{10} x + 576 \, a^{3} b^{2} c^{4} d^{10} x - 3840 \, a^{4} c^{5} d^{10} x + b^{9} d^{10} + 18 \, a b^{7} c d^{10} - 312 \, a^{2} b^{5} c^{2} d^{10} + 1376 \, a^{3} b^{3} c^{3} d^{10} - 1920 \, a^{4} b c^{4} d^{10}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} + \frac{128 \,{\left (8 \, c^{22} d^{10} x^{5} + 20 \, b c^{21} d^{10} x^{4} + 30 \, b^{2} c^{20} d^{10} x^{3} - 40 \, a c^{21} d^{10} x^{3} + 25 \, b^{3} c^{19} d^{10} x^{2} - 60 \, a b c^{20} d^{10} x^{2} + 25 \, b^{4} c^{18} d^{10} x - 150 \, a b^{2} c^{19} d^{10} x + 240 \, a^{2} c^{20} d^{10} x\right )}}{5 \, c^{15}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^10/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

252*(b^6*c^2*d^10 - 12*a*b^4*c^3*d^10 + 48*a^2*b^2*c^4*d^10 - 64*a^3*c^5*d^10)*arctan((2*c*x + b)/sqrt(-b^2 +
4*a*c))/sqrt(-b^2 + 4*a*c) - 1/2*(68*b^6*c^3*d^10*x^3 - 816*a*b^4*c^4*d^10*x^3 + 3264*a^2*b^2*c^5*d^10*x^3 - 4
352*a^3*c^6*d^10*x^3 + 102*b^7*c^2*d^10*x^2 - 1224*a*b^5*c^3*d^10*x^2 + 4896*a^2*b^3*c^4*d^10*x^2 - 6528*a^3*b
*c^5*d^10*x^2 + 36*b^8*c*d^10*x - 372*a*b^6*c^2*d^10*x + 1008*a^2*b^4*c^3*d^10*x + 576*a^3*b^2*c^4*d^10*x - 38
40*a^4*c^5*d^10*x + b^9*d^10 + 18*a*b^7*c*d^10 - 312*a^2*b^5*c^2*d^10 + 1376*a^3*b^3*c^3*d^10 - 1920*a^4*b*c^4
*d^10)/(c*x^2 + b*x + a)^2 + 128/5*(8*c^22*d^10*x^5 + 20*b*c^21*d^10*x^4 + 30*b^2*c^20*d^10*x^3 - 40*a*c^21*d^
10*x^3 + 25*b^3*c^19*d^10*x^2 - 60*a*b*c^20*d^10*x^2 + 25*b^4*c^18*d^10*x - 150*a*b^2*c^19*d^10*x + 240*a^2*c^
20*d^10*x)/c^15