\(\int \frac {(b d+2 c d x)^6}{(a+b x+c x^2)^{5/2}} \, dx\) [1250]

Optimal result

Integrand size = 26, antiderivative size = 136 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}+80 c^2 d^6 (b+2 c x) \sqrt {a+b x+c x^2}+40 c^{3/2} \left (b^2-4 a c\right ) d^6 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]

[Out]

-2/3*d^6*(2*c*x+b)^5/(c*x^2+b*x+a)^(3/2)+40*c^(3/2)*(-4*a*c+b^2)*d^6*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+
a)^(1/2))-40/3*c*d^6*(2*c*x+b)^3/(c*x^2+b*x+a)^(1/2)+80*c^2*d^6*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {700, 706, 635, 212} \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^{5/2}} \, dx=40 c^{3/2} d^6 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )+80 c^2 d^6 (b+2 c x) \sqrt {a+b x+c x^2}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}-\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[In]

Int[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*d^6*(b + 2*c*x)^5)/(3*(a + b*x + c*x^2)^(3/2)) - (40*c*d^6*(b + 2*c*x)^3)/(3*Sqrt[a + b*x + c*x^2]) + 80*c
^2*d^6*(b + 2*c*x)*Sqrt[a + b*x + c*x^2] + 40*c^(3/2)*(b^2 - 4*a*c)*d^6*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 700

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 706

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{3} \left (20 c d^2\right ) \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx \\ & = -\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}+\left (80 c^2 d^4\right ) \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}+80 c^2 d^6 (b+2 c x) \sqrt {a+b x+c x^2}+\left (40 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}+80 c^2 d^6 (b+2 c x) \sqrt {a+b x+c x^2}+\left (80 c^2 \left (b^2-4 a c\right ) d^6\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right ) \\ & = -\frac {2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {40 c d^6 (b+2 c x)^3}{3 \sqrt {a+b x+c x^2}}+80 c^2 d^6 (b+2 c x) \sqrt {a+b x+c x^2}+40 c^{3/2} \left (b^2-4 a c\right ) d^6 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.07 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^{5/2}} \, dx=d^6 \left (-\frac {2 (b+2 c x) \left (b^4+28 b^3 c x+4 b^2 c \left (5 a+c x^2\right )-16 b c^2 x \left (10 a+3 c x^2\right )-8 c^2 \left (15 a^2+20 a c x^2+3 c^2 x^4\right )\right )}{3 (a+x (b+c x))^{3/2}}-80 c^{3/2} \left (-b^2+4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )\right ) \]

[In]

Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^(5/2),x]

[Out]

d^6*((-2*(b + 2*c*x)*(b^4 + 28*b^3*c*x + 4*b^2*c*(5*a + c*x^2) - 16*b*c^2*x*(10*a + 3*c*x^2) - 8*c^2*(15*a^2 +
20*a*c*x^2 + 3*c^2*x^4)))/(3*(a + x*(b + c*x))^(3/2)) - 80*c^(3/2)*(-b^2 + 4*a*c)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[
a] + Sqrt[a + x*(b + c*x)])])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(711\) vs. \(2(118)=236\).

Time = 3.12 (sec) , antiderivative size = 712, normalized size of antiderivative = 5.24

method	result	size
risch	\(16 c^{2} d^{6} \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}+\left (-40 c^{\frac {3}{2}} \left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )+\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \left (-\frac {2 \sqrt {{\left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}^{2} c +\sqrt {-4 a c +b^{2}}\, \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{3 \sqrt {-4 a c +b^{2}}\, {\left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}^{2}}+\frac {4 c \sqrt {{\left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}^{2} c +\sqrt {-4 a c +b^{2}}\, \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{3 \left (-4 a c +b^{2}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\right )+\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \left (\frac {2 \sqrt {{\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}^{2} c -\sqrt {-4 a c +b^{2}}\, \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{3 \sqrt {-4 a c +b^{2}}\, {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}^{2}}+\frac {4 c \sqrt {{\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}^{2} c -\sqrt {-4 a c +b^{2}}\, \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{3 \left (-4 a c +b^{2}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\right )-\frac {20 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {{\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}^{2} c -\sqrt {-4 a c +b^{2}}\, \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}-\frac {20 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {{\left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}^{2} c +\sqrt {-4 a c +b^{2}}\, \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\right ) d^{6}\)	\(712\)
default	\(\text {Expression too large to display}\)	\(3217\)

[In]

int((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

16*c^2*d^6*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)+(-40*c^(3/2)*(4*a*c-b^2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+
(16*a^2*c^2-8*a*b^2*c+b^4)*(-2/3/(-4*a*c+b^2)^(1/2)/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))^2*((x-1/2/c*(-b+(-4*a*c+
b^2)^(1/2)))^2*c+(-4*a*c+b^2)^(1/2)*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)+4/3*c/(-4*a*c+b^2)/(x-1/2/c*(-b+(
-4*a*c+b^2)^(1/2)))*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))^2*c+(-4*a*c+b^2)^(1/2)*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))
))^(1/2))+(16*a^2*c^2-8*a*b^2*c+b^4)*(2/3/(-4*a*c+b^2)^(1/2)/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*((x+1/2*(b+(-4
*a*c+b^2)^(1/2))/c)^2*c-(-4*a*c+b^2)^(1/2)*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)+4/3*c/(-4*a*c+b^2)/(x+1/2*(
b+(-4*a*c+b^2)^(1/2))/c)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*c-(-4*a*c+b^2)^(1/2)*(x+1/2*(b+(-4*a*c+b^2)^(1/2)
)/c))^(1/2))-20*c*(16*a^2*c^2-8*a*b^2*c+b^4)/(-4*a*c+b^2)/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+1/2*(b+(-4*a*c+
b^2)^(1/2))/c)^2*c-(-4*a*c+b^2)^(1/2)*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)-20*c*(16*a^2*c^2-8*a*b^2*c+b^4)/
(-4*a*c+b^2)/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))^2*c+(-4*a*c+b^2)^(1/2)*(x-1/
2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))*d^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (118) = 236\).

Time = 0.67 (sec) , antiderivative size = 693, normalized size of antiderivative = 5.10 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\left [-\frac {2 \, {\left (30 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{4} + 2 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{3} + {\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x^{2} + 2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{6} x + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} d^{6}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - {\left (48 \, c^{5} d^{6} x^{5} + 120 \, b c^{4} d^{6} x^{4} + 40 \, {\left (b^{2} c^{3} + 8 \, a c^{4}\right )} d^{6} x^{3} - 60 \, {\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{6} x^{2} - 30 \, {\left (b^{4} c - 4 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x - {\left (b^{5} + 20 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\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 {2 \, {\left (60 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{4} + 2 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{3} + {\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x^{2} + 2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{6} x + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} d^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - {\left (48 \, c^{5} d^{6} x^{5} + 120 \, b c^{4} d^{6} x^{4} + 40 \, {\left (b^{2} c^{3} + 8 \, a c^{4}\right )} d^{6} x^{3} - 60 \, {\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{6} x^{2} - 30 \, {\left (b^{4} c - 4 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x - {\left (b^{5} + 20 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\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 ] \]

[In]

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-2/3*(30*((b^2*c^3 - 4*a*c^4)*d^6*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*d^6*x^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d
^6*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*d^6*x + (a^2*b^2*c - 4*a^3*c^2)*d^6)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2
+ 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - (48*c^5*d^6*x^5 + 120*b*c^4*d^6*x^4 + 40*(b^2*c^3 +
8*a*c^4)*d^6*x^3 - 60*(b^3*c^2 - 8*a*b*c^3)*d^6*x^2 - 30*(b^4*c - 4*a*b^2*c^2 - 8*a^2*c^3)*d^6*x - (b^5 + 20*a
*b^3*c - 120*a^2*b*c^2)*d^6)*sqrt(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),
-2/3*(60*((b^2*c^3 - 4*a*c^4)*d^6*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*d^6*x^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d
^6*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*d^6*x + (a^2*b^2*c - 4*a^3*c^2)*d^6)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x +
a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - (48*c^5*d^6*x^5 + 120*b*c^4*d^6*x^4 + 40*(b^2*c^3 + 8*a*c^
4)*d^6*x^3 - 60*(b^3*c^2 - 8*a*b*c^3)*d^6*x^2 - 30*(b^4*c - 4*a*b^2*c^2 - 8*a^2*c^3)*d^6*x - (b^5 + 20*a*b^3*c
- 120*a^2*b*c^2)*d^6)*sqrt(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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((2*c*d*x+b*d)**6/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^(5/2),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 deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (118) = 236\).

Time = 0.30 (sec) , antiderivative size = 526, normalized size of antiderivative = 3.87 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {40 \, {\left (b^{2} c^{2} d^{6} - 4 \, a c^{3} d^{6}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{\sqrt {c}} + \frac {2 \, {\left (2 \, {\left (2 \, {\left (2 \, {\left (3 \, {\left (\frac {2 \, {\left (b^{4} c^{8} d^{6} - 8 \, a b^{2} c^{9} d^{6} + 16 \, a^{2} c^{10} d^{6}\right )} x}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}} + \frac {5 \, {\left (b^{5} c^{7} d^{6} - 8 \, a b^{3} c^{8} d^{6} + 16 \, a^{2} b c^{9} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x + \frac {5 \, {\left (b^{6} c^{6} d^{6} - 48 \, a^{2} b^{2} c^{8} d^{6} + 128 \, a^{3} c^{9} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {15 \, {\left (b^{7} c^{5} d^{6} - 16 \, a b^{5} c^{6} d^{6} + 80 \, a^{2} b^{3} c^{7} d^{6} - 128 \, a^{3} b c^{8} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {15 \, {\left (b^{8} c^{4} d^{6} - 12 \, a b^{6} c^{5} d^{6} + 40 \, a^{2} b^{4} c^{6} d^{6} - 128 \, a^{4} c^{8} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {b^{9} c^{3} d^{6} + 12 \, a b^{7} c^{4} d^{6} - 264 \, a^{2} b^{5} c^{5} d^{6} + 1280 \, a^{3} b^{3} c^{6} d^{6} - 1920 \, a^{4} b c^{7} d^{6}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]

[In]

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

-40*(b^2*c^2*d^6 - 4*a*c^3*d^6)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/sqrt(c) + 2/3*(2*(
2*(2*(3*(2*(b^4*c^8*d^6 - 8*a*b^2*c^9*d^6 + 16*a^2*c^10*d^6)*x/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5) + 5*(b^5*c
^7*d^6 - 8*a*b^3*c^8*d^6 + 16*a^2*b*c^9*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x + 5*(b^6*c^6*d^6 - 48*a^2
*b^2*c^8*d^6 + 128*a^3*c^9*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - 15*(b^7*c^5*d^6 - 16*a*b^5*c^6*d^6 +
80*a^2*b^3*c^7*d^6 - 128*a^3*b*c^8*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - 15*(b^8*c^4*d^6 - 12*a*b^6*
c^5*d^6 + 40*a^2*b^4*c^6*d^6 - 128*a^4*c^8*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - (b^9*c^3*d^6 + 12*a*
b^7*c^4*d^6 - 264*a^2*b^5*c^5*d^6 + 1280*a^3*b^3*c^6*d^6 - 1920*a^4*b*c^7*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2
*c^5))/(c*x^2 + b*x + a)^(3/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^6}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]

[In]

int((b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^(5/2),x)

[Out]

int((b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^(5/2), x)