3.11.2 \(\int \frac {1}{(b d+2 c d x)^4 (a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=168 \[ \frac {140 c^2}{d^4 \left (b^2-4 a c\right )^4 (b+2 c x)}+\frac {140 c^2}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3}-\frac {140 c^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{d^4 \left (b^2-4 a c\right )^{9/2}}+\frac {7 c}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \left (a+b x+c x^2\right )}-\frac {1}{2 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^2} \]

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Rubi [A]  time = 0.15, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {687, 693, 618, 206} \begin {gather*} \frac {140 c^2}{d^4 \left (b^2-4 a c\right )^4 (b+2 c x)}+\frac {140 c^2}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3}-\frac {140 c^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{d^4 \left (b^2-4 a c\right )^{9/2}}+\frac {7 c}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \left (a+b x+c x^2\right )}-\frac {1}{2 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3),x]

[Out]

(140*c^2)/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3) + (140*c^2)/((b^2 - 4*a*c)^4*d^4*(b + 2*c*x)) - 1/(2*(b^2 - 4*
a*c)*d^4*(b + 2*c*x)^3*(a + b*x + c*x^2)^2) + (7*c)/((b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^3*(a + b*x + c*x^2)) - (1
40*c^2*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(9/2)*d^4)

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])

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 687

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

Rule 693

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-2*b*d*(d + e*x)^(m
+ 1)*(a + b*x + c*x^2)^(p + 1))/(d^2*(m + 1)*(b^2 - 4*a*c)), x] + Dist[(b^2*(m + 2*p + 3))/(d^2*(m + 1)*(b^2
- 4*a*c)), Int[(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] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

Rubi steps

\begin {align*} \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^3} \, dx &=-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}-\frac {(7 c) \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^2} \, dx}{b^2-4 a c}\\ &=-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}+\frac {7 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )}+\frac {\left (70 c^2\right ) \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2}\\ &=\frac {140 c^2}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}+\frac {7 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )}+\frac {\left (70 c^2\right ) \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^3 d^2}\\ &=\frac {140 c^2}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac {140 c^2}{\left (b^2-4 a c\right )^4 d^4 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}+\frac {7 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )}+\frac {\left (70 c^2\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4 d^4}\\ &=\frac {140 c^2}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac {140 c^2}{\left (b^2-4 a c\right )^4 d^4 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}+\frac {7 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )}-\frac {\left (140 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^4 d^4}\\ &=\frac {140 c^2}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac {140 c^2}{\left (b^2-4 a c\right )^4 d^4 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}+\frac {7 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )}-\frac {140 c^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2} d^4}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 140, normalized size = 0.83 \begin {gather*} \frac {\frac {64 c^2 \left (b^2-4 a c\right )}{(b+2 c x)^3}+\frac {840 c^2 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac {66 c (b+2 c x)}{a+x (b+c x)}+\frac {576 c^2}{b+2 c x}}{6 d^4 \left (b^2-4 a c\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3),x]

[Out]

((64*c^2*(b^2 - 4*a*c))/(b + 2*c*x)^3 + (576*c^2)/(b + 2*c*x) - (3*(b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x)
)^2 + (66*c*(b + 2*c*x))/(a + x*(b + c*x)) + (840*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*
c])/(6*(b^2 - 4*a*c)^4*d^4)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3),x]

[Out]

IntegrateAlgebraic[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3), x]

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fricas [B]  time = 0.45, size = 2077, normalized size = 12.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[-1/6*(3*b^8 - 90*a*b^6*c - 328*a^2*b^4*c^2 + 2816*a^3*b^2*c^3 - 1024*a^4*c^4 - 3360*(b^2*c^6 - 4*a*c^7)*x^6 -
10080*(b^3*c^5 - 4*a*b*c^6)*x^5 - 5600*(2*b^4*c^4 - 7*a*b^2*c^5 - 4*a^2*c^6)*x^4 - 5600*(b^5*c^3 - 2*a*b^3*c^
4 - 8*a^2*b*c^5)*x^3 - 14*(83*b^6*c^2 + 204*a*b^4*c^3 - 2016*a^2*b^2*c^4 - 512*a^3*c^5)*x^2 - 420*(8*c^7*x^7 +
28*b*c^6*x^6 + a^2*b^3*c^2 + 2*(19*b^2*c^5 + 8*a*c^6)*x^5 + 5*(5*b^3*c^4 + 8*a*b*c^5)*x^4 + 4*(2*b^4*c^3 + 9*
a*b^2*c^4 + 2*a^2*c^5)*x^3 + (b^5*c^2 + 14*a*b^3*c^3 + 12*a^2*b*c^4)*x^2 + 2*(a*b^4*c^2 + 3*a^2*b^2*c^3)*x)*sq
rt(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)) - 1
4*(3*b^7*c + 124*a*b^5*c^2 - 416*a^2*b^3*c^3 - 512*a^3*b*c^4)*x)/(8*(b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7
- 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10)*d^4*x^7 + 28*(b^11*c^4 - 20*a*b^9*c^5 + 160*a^2*b^7*c^6
- 640*a^3*b^5*c^7 + 1280*a^4*b^3*c^8 - 1024*a^5*b*c^9)*d^4*x^6 + 2*(19*b^12*c^3 - 372*a*b^10*c^4 + 2880*a^2*b
^8*c^5 - 10880*a^3*b^6*c^6 + 19200*a^4*b^4*c^7 - 9216*a^5*b^2*c^8 - 8192*a^6*c^9)*d^4*x^5 + 5*(5*b^13*c^2 - 92
*a*b^11*c^3 + 640*a^2*b^9*c^4 - 1920*a^3*b^7*c^5 + 1280*a^4*b^5*c^6 + 5120*a^5*b^3*c^7 - 8192*a^6*b*c^8)*d^4*x
^4 + 4*(2*b^14*c - 31*a*b^12*c^2 + 142*a^2*b^10*c^3 + 120*a^3*b^8*c^4 - 2880*a^4*b^6*c^5 + 8192*a^5*b^4*c^6 -
6656*a^6*b^2*c^7 - 2048*a^7*c^8)*d^4*x^3 + (b^15 - 6*a*b^13*c - 108*a^2*b^11*c^2 + 1360*a^3*b^9*c^3 - 5760*a^4
*b^7*c^4 + 9216*a^5*b^5*c^5 + 1024*a^6*b^3*c^6 - 12288*a^7*b*c^7)*d^4*x^2 + 2*(a*b^14 - 17*a^2*b^12*c + 100*a^
3*b^10*c^2 - 160*a^4*b^8*c^3 - 640*a^5*b^6*c^4 + 2816*a^6*b^4*c^5 - 3072*a^7*b^2*c^6)*d^4*x + (a^2*b^13 - 20*a
^3*b^11*c + 160*a^4*b^9*c^2 - 640*a^5*b^7*c^3 + 1280*a^6*b^5*c^4 - 1024*a^7*b^3*c^5)*d^4), -1/6*(3*b^8 - 90*a*
b^6*c - 328*a^2*b^4*c^2 + 2816*a^3*b^2*c^3 - 1024*a^4*c^4 - 3360*(b^2*c^6 - 4*a*c^7)*x^6 - 10080*(b^3*c^5 - 4*
a*b*c^6)*x^5 - 5600*(2*b^4*c^4 - 7*a*b^2*c^5 - 4*a^2*c^6)*x^4 - 5600*(b^5*c^3 - 2*a*b^3*c^4 - 8*a^2*b*c^5)*x^3
- 14*(83*b^6*c^2 + 204*a*b^4*c^3 - 2016*a^2*b^2*c^4 - 512*a^3*c^5)*x^2 + 840*(8*c^7*x^7 + 28*b*c^6*x^6 + a^2*
b^3*c^2 + 2*(19*b^2*c^5 + 8*a*c^6)*x^5 + 5*(5*b^3*c^4 + 8*a*b*c^5)*x^4 + 4*(2*b^4*c^3 + 9*a*b^2*c^4 + 2*a^2*c^
5)*x^3 + (b^5*c^2 + 14*a*b^3*c^3 + 12*a^2*b*c^4)*x^2 + 2*(a*b^4*c^2 + 3*a^2*b^2*c^3)*x)*sqrt(-b^2 + 4*a*c)*arc
tan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 14*(3*b^7*c + 124*a*b^5*c^2 - 416*a^2*b^3*c^3 - 512*a^3*b
*c^4)*x)/(8*(b^10*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10)*d
^4*x^7 + 28*(b^11*c^4 - 20*a*b^9*c^5 + 160*a^2*b^7*c^6 - 640*a^3*b^5*c^7 + 1280*a^4*b^3*c^8 - 1024*a^5*b*c^9)*
d^4*x^6 + 2*(19*b^12*c^3 - 372*a*b^10*c^4 + 2880*a^2*b^8*c^5 - 10880*a^3*b^6*c^6 + 19200*a^4*b^4*c^7 - 9216*a^
5*b^2*c^8 - 8192*a^6*c^9)*d^4*x^5 + 5*(5*b^13*c^2 - 92*a*b^11*c^3 + 640*a^2*b^9*c^4 - 1920*a^3*b^7*c^5 + 1280*
a^4*b^5*c^6 + 5120*a^5*b^3*c^7 - 8192*a^6*b*c^8)*d^4*x^4 + 4*(2*b^14*c - 31*a*b^12*c^2 + 142*a^2*b^10*c^3 + 12
0*a^3*b^8*c^4 - 2880*a^4*b^6*c^5 + 8192*a^5*b^4*c^6 - 6656*a^6*b^2*c^7 - 2048*a^7*c^8)*d^4*x^3 + (b^15 - 6*a*b
^13*c - 108*a^2*b^11*c^2 + 1360*a^3*b^9*c^3 - 5760*a^4*b^7*c^4 + 9216*a^5*b^5*c^5 + 1024*a^6*b^3*c^6 - 12288*a
^7*b*c^7)*d^4*x^2 + 2*(a*b^14 - 17*a^2*b^12*c + 100*a^3*b^10*c^2 - 160*a^4*b^8*c^3 - 640*a^5*b^6*c^4 + 2816*a^
6*b^4*c^5 - 3072*a^7*b^2*c^6)*d^4*x + (a^2*b^13 - 20*a^3*b^11*c + 160*a^4*b^9*c^2 - 640*a^5*b^7*c^3 + 1280*a^6
*b^5*c^4 - 1024*a^7*b^3*c^5)*d^4)]

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giac [A]  time = 0.18, size = 311, normalized size = 1.85 \begin {gather*} \frac {140 \, c^{2} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {44 \, c^{3} x^{3} + 66 \, b c^{2} x^{2} + 20 \, b^{2} c x + 52 \, a c^{2} x - b^{3} + 26 \, a b c}{2 \, {\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )} {\left (c x^{2} + b x + a\right )}^{2}} + \frac {64 \, {\left (18 \, c^{4} x^{2} + 18 \, b c^{3} x + 5 \, b^{2} c^{2} - 2 \, a c^{3}\right )}}{3 \, {\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )} {\left (2 \, c x + b\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

140*c^2*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^8*d^4 - 16*a*b^6*c*d^4 + 96*a^2*b^4*c^2*d^4 - 256*a^3*b^2*c
^3*d^4 + 256*a^4*c^4*d^4)*sqrt(-b^2 + 4*a*c)) + 1/2*(44*c^3*x^3 + 66*b*c^2*x^2 + 20*b^2*c*x + 52*a*c^2*x - b^3
+ 26*a*b*c)/((b^8*d^4 - 16*a*b^6*c*d^4 + 96*a^2*b^4*c^2*d^4 - 256*a^3*b^2*c^3*d^4 + 256*a^4*c^4*d^4)*(c*x^2 +
b*x + a)^2) + 64/3*(18*c^4*x^2 + 18*b*c^3*x + 5*b^2*c^2 - 2*a*c^3)/((b^8*d^4 - 16*a*b^6*c*d^4 + 96*a^2*b^4*c^
2*d^4 - 256*a^3*b^2*c^3*d^4 + 256*a^4*c^4*d^4)*(2*c*x + b)^3)

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maple [A]  time = 0.07, size = 301, normalized size = 1.79 \begin {gather*} \frac {22 c^{3} x^{3}}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{4}}+\frac {33 b \,c^{2} x^{2}}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{4}}+\frac {26 a \,c^{2} x}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{4}}+\frac {10 b^{2} c x}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{4}}+\frac {13 a b c}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{4}}-\frac {b^{3}}{2 \left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{4}}+\frac {140 c^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {9}{2}} d^{4}}+\frac {96 c^{2}}{\left (4 a c -b^{2}\right )^{4} \left (2 c x +b \right ) d^{4}}-\frac {32 c^{2}}{3 \left (4 a c -b^{2}\right )^{3} \left (2 c x +b \right )^{3} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^3,x)

[Out]

96/d^4/(4*a*c-b^2)^4*c^2/(2*c*x+b)-32/3/d^4*c^2/(4*a*c-b^2)^3/(2*c*x+b)^3+22/d^4/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2
*c^3*x^3+33/d^4/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*b*c^2*x^2+26/d^4/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*a*c^2*x+10/d^4/(4
*a*c-b^2)^4/(c*x^2+b*x+a)^2*b^2*c*x+13/d^4/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*a*b*c-1/2/d^4/(4*a*c-b^2)^4/(c*x^2+b*
x+a)^2*b^3+140/d^4/(4*a*c-b^2)^(9/2)*c^2*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 1.36, size = 827, normalized size = 4.92 \begin {gather*} \frac {140\,c^2\,\mathrm {atan}\left (\frac {\frac {70\,c^2\,\left (256\,a^4\,b\,c^4\,d^4-256\,a^3\,b^3\,c^3\,d^4+96\,a^2\,b^5\,c^2\,d^4-16\,a\,b^7\,c\,d^4+b^9\,d^4\right )}{d^4\,{\left (4\,a\,c-b^2\right )}^{9/2}}+\frac {140\,c^3\,x\,\left (256\,a^4\,c^4\,d^4-256\,a^3\,b^2\,c^3\,d^4+96\,a^2\,b^4\,c^2\,d^4-16\,a\,b^6\,c\,d^4+b^8\,d^4\right )}{d^4\,{\left (4\,a\,c-b^2\right )}^{9/2}}}{70\,c^2}\right )}{d^4\,{\left (4\,a\,c-b^2\right )}^{9/2}}-\frac {\frac {2800\,x^3\,\left (b^3\,c^3+2\,a\,b\,c^4\right )}{3\,\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}-\frac {256\,a^3\,c^3-640\,a^2\,b^2\,c^2-78\,a\,b^4\,c+3\,b^6}{6\,\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {7\,x^2\,\left (128\,a^2\,c^4+536\,a\,b^2\,c^3+83\,b^4\,c^2\right )}{3\,\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {560\,c^6\,x^6}{\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {7\,b\,x\,\left (128\,a^2\,c^3+136\,a\,b^2\,c^2+3\,b^4\,c\right )}{3\,\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {1680\,b\,c^5\,x^5}{\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {2800\,c\,x^4\,\left (2\,b^2\,c^3+a\,c^4\right )}{3\,\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}}{x^2\,\left (12\,a^2\,b\,c^2\,d^4+14\,a\,b^3\,c\,d^4+b^5\,d^4\right )+x^5\,\left (38\,b^2\,c^3\,d^4+16\,a\,c^4\,d^4\right )+x\,\left (6\,c\,a^2\,b^2\,d^4+2\,a\,b^4\,d^4\right )+x^3\,\left (8\,a^2\,c^3\,d^4+36\,a\,b^2\,c^2\,d^4+8\,b^4\,c\,d^4\right )+x^4\,\left (25\,b^3\,c^2\,d^4+40\,a\,b\,c^3\,d^4\right )+a^2\,b^3\,d^4+8\,c^5\,d^4\,x^7+28\,b\,c^4\,d^4\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3),x)

[Out]

(140*c^2*atan(((70*c^2*(b^9*d^4 + 256*a^4*b*c^4*d^4 + 96*a^2*b^5*c^2*d^4 - 256*a^3*b^3*c^3*d^4 - 16*a*b^7*c*d^
4))/(d^4*(4*a*c - b^2)^(9/2)) + (140*c^3*x*(b^8*d^4 + 256*a^4*c^4*d^4 + 96*a^2*b^4*c^2*d^4 - 256*a^3*b^2*c^3*d
^4 - 16*a*b^6*c*d^4))/(d^4*(4*a*c - b^2)^(9/2)))/(70*c^2)))/(d^4*(4*a*c - b^2)^(9/2)) - ((2800*x^3*(b^3*c^3 +
2*a*b*c^4))/(3*(4*a*c - b^2)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - (3*b^6 + 256*a^3*c^3 - 640*a^
2*b^2*c^2 - 78*a*b^4*c)/(6*(4*a*c - b^2)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (7*x^2*(128*a^2*c
^4 + 83*b^4*c^2 + 536*a*b^2*c^3))/(3*(4*a*c - b^2)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (560*c^
6*x^6)/((4*a*c - b^2)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (7*b*x*(3*b^4*c + 128*a^2*c^3 + 136*
a*b^2*c^2))/(3*(4*a*c - b^2)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (1680*b*c^5*x^5)/((4*a*c - b^
2)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (2800*c*x^4*(a*c^4 + 2*b^2*c^3))/(3*(4*a*c - b^2)*(b^6
- 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)))/(x^2*(b^5*d^4 + 12*a^2*b*c^2*d^4 + 14*a*b^3*c*d^4) + x^5*(16*a*c
^4*d^4 + 38*b^2*c^3*d^4) + x*(2*a*b^4*d^4 + 6*a^2*b^2*c*d^4) + x^3*(8*b^4*c*d^4 + 8*a^2*c^3*d^4 + 36*a*b^2*c^2
*d^4) + x^4*(25*b^3*c^2*d^4 + 40*a*b*c^3*d^4) + a^2*b^3*d^4 + 8*c^5*d^4*x^7 + 28*b*c^4*d^4*x^6)

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sympy [B]  time = 6.63, size = 1238, normalized size = 7.37 \begin {gather*} - \frac {70 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} \log {\left (x + \frac {- 71680 a^{5} c^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 89600 a^{4} b^{2} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 44800 a^{3} b^{4} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 11200 a^{2} b^{6} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 1400 a b^{8} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 70 b^{10} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 70 b c^{2}}{140 c^{3}} \right )}}{d^{4}} + \frac {70 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} \log {\left (x + \frac {71680 a^{5} c^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 89600 a^{4} b^{2} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 44800 a^{3} b^{4} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 11200 a^{2} b^{6} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 1400 a b^{8} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 70 b^{10} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 70 b c^{2}}{140 c^{3}} \right )}}{d^{4}} + \frac {- 256 a^{3} c^{3} + 640 a^{2} b^{2} c^{2} + 78 a b^{4} c - 3 b^{6} + 10080 b c^{5} x^{5} + 3360 c^{6} x^{6} + x^{4} \left (5600 a c^{5} + 11200 b^{2} c^{4}\right ) + x^{3} \left (11200 a b c^{4} + 5600 b^{3} c^{3}\right ) + x^{2} \left (1792 a^{2} c^{4} + 7504 a b^{2} c^{3} + 1162 b^{4} c^{2}\right ) + x \left (1792 a^{2} b c^{3} + 1904 a b^{3} c^{2} + 42 b^{5} c\right )}{1536 a^{6} b^{3} c^{4} d^{4} - 1536 a^{5} b^{5} c^{3} d^{4} + 576 a^{4} b^{7} c^{2} d^{4} - 96 a^{3} b^{9} c d^{4} + 6 a^{2} b^{11} d^{4} + x^{7} \left (12288 a^{4} c^{9} d^{4} - 12288 a^{3} b^{2} c^{8} d^{4} + 4608 a^{2} b^{4} c^{7} d^{4} - 768 a b^{6} c^{6} d^{4} + 48 b^{8} c^{5} d^{4}\right ) + x^{6} \left (43008 a^{4} b c^{8} d^{4} - 43008 a^{3} b^{3} c^{7} d^{4} + 16128 a^{2} b^{5} c^{6} d^{4} - 2688 a b^{7} c^{5} d^{4} + 168 b^{9} c^{4} d^{4}\right ) + x^{5} \left (24576 a^{5} c^{8} d^{4} + 33792 a^{4} b^{2} c^{7} d^{4} - 49152 a^{3} b^{4} c^{6} d^{4} + 20352 a^{2} b^{6} c^{5} d^{4} - 3552 a b^{8} c^{4} d^{4} + 228 b^{10} c^{3} d^{4}\right ) + x^{4} \left (61440 a^{5} b c^{7} d^{4} - 23040 a^{4} b^{3} c^{6} d^{4} - 15360 a^{3} b^{5} c^{5} d^{4} + 10560 a^{2} b^{7} c^{4} d^{4} - 2160 a b^{9} c^{3} d^{4} + 150 b^{11} c^{2} d^{4}\right ) + x^{3} \left (12288 a^{6} c^{7} d^{4} + 43008 a^{5} b^{2} c^{6} d^{4} - 38400 a^{4} b^{4} c^{5} d^{4} + 7680 a^{3} b^{6} c^{4} d^{4} + 1200 a^{2} b^{8} c^{3} d^{4} - 552 a b^{10} c^{2} d^{4} + 48 b^{12} c d^{4}\right ) + x^{2} \left (18432 a^{6} b c^{6} d^{4} + 3072 a^{5} b^{3} c^{5} d^{4} - 13056 a^{4} b^{5} c^{4} d^{4} + 5376 a^{3} b^{7} c^{3} d^{4} - 696 a^{2} b^{9} c^{2} d^{4} - 12 a b^{11} c d^{4} + 6 b^{13} d^{4}\right ) + x \left (9216 a^{6} b^{2} c^{5} d^{4} - 6144 a^{5} b^{4} c^{4} d^{4} + 384 a^{4} b^{6} c^{3} d^{4} + 576 a^{3} b^{8} c^{2} d^{4} - 156 a^{2} b^{10} c d^{4} + 12 a b^{12} d^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a)**3,x)

[Out]

-70*c**2*sqrt(-1/(4*a*c - b**2)**9)*log(x + (-71680*a**5*c**7*sqrt(-1/(4*a*c - b**2)**9) + 89600*a**4*b**2*c**
6*sqrt(-1/(4*a*c - b**2)**9) - 44800*a**3*b**4*c**5*sqrt(-1/(4*a*c - b**2)**9) + 11200*a**2*b**6*c**4*sqrt(-1/
(4*a*c - b**2)**9) - 1400*a*b**8*c**3*sqrt(-1/(4*a*c - b**2)**9) + 70*b**10*c**2*sqrt(-1/(4*a*c - b**2)**9) +
70*b*c**2)/(140*c**3))/d**4 + 70*c**2*sqrt(-1/(4*a*c - b**2)**9)*log(x + (71680*a**5*c**7*sqrt(-1/(4*a*c - b**
2)**9) - 89600*a**4*b**2*c**6*sqrt(-1/(4*a*c - b**2)**9) + 44800*a**3*b**4*c**5*sqrt(-1/(4*a*c - b**2)**9) - 1
1200*a**2*b**6*c**4*sqrt(-1/(4*a*c - b**2)**9) + 1400*a*b**8*c**3*sqrt(-1/(4*a*c - b**2)**9) - 70*b**10*c**2*s
qrt(-1/(4*a*c - b**2)**9) + 70*b*c**2)/(140*c**3))/d**4 + (-256*a**3*c**3 + 640*a**2*b**2*c**2 + 78*a*b**4*c -
3*b**6 + 10080*b*c**5*x**5 + 3360*c**6*x**6 + x**4*(5600*a*c**5 + 11200*b**2*c**4) + x**3*(11200*a*b*c**4 + 5
600*b**3*c**3) + x**2*(1792*a**2*c**4 + 7504*a*b**2*c**3 + 1162*b**4*c**2) + x*(1792*a**2*b*c**3 + 1904*a*b**3
*c**2 + 42*b**5*c))/(1536*a**6*b**3*c**4*d**4 - 1536*a**5*b**5*c**3*d**4 + 576*a**4*b**7*c**2*d**4 - 96*a**3*b
**9*c*d**4 + 6*a**2*b**11*d**4 + x**7*(12288*a**4*c**9*d**4 - 12288*a**3*b**2*c**8*d**4 + 4608*a**2*b**4*c**7*
d**4 - 768*a*b**6*c**6*d**4 + 48*b**8*c**5*d**4) + x**6*(43008*a**4*b*c**8*d**4 - 43008*a**3*b**3*c**7*d**4 +
16128*a**2*b**5*c**6*d**4 - 2688*a*b**7*c**5*d**4 + 168*b**9*c**4*d**4) + x**5*(24576*a**5*c**8*d**4 + 33792*a
**4*b**2*c**7*d**4 - 49152*a**3*b**4*c**6*d**4 + 20352*a**2*b**6*c**5*d**4 - 3552*a*b**8*c**4*d**4 + 228*b**10
*c**3*d**4) + x**4*(61440*a**5*b*c**7*d**4 - 23040*a**4*b**3*c**6*d**4 - 15360*a**3*b**5*c**5*d**4 + 10560*a**
2*b**7*c**4*d**4 - 2160*a*b**9*c**3*d**4 + 150*b**11*c**2*d**4) + x**3*(12288*a**6*c**7*d**4 + 43008*a**5*b**2
*c**6*d**4 - 38400*a**4*b**4*c**5*d**4 + 7680*a**3*b**6*c**4*d**4 + 1200*a**2*b**8*c**3*d**4 - 552*a*b**10*c**
2*d**4 + 48*b**12*c*d**4) + x**2*(18432*a**6*b*c**6*d**4 + 3072*a**5*b**3*c**5*d**4 - 13056*a**4*b**5*c**4*d**
4 + 5376*a**3*b**7*c**3*d**4 - 696*a**2*b**9*c**2*d**4 - 12*a*b**11*c*d**4 + 6*b**13*d**4) + x*(9216*a**6*b**2
*c**5*d**4 - 6144*a**5*b**4*c**4*d**4 + 384*a**4*b**6*c**3*d**4 + 576*a**3*b**8*c**2*d**4 - 156*a**2*b**10*c*d
**4 + 12*a*b**12*d**4))

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