\(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^6} \, dx\) [1227]

Optimal result

Integrand size = 26, antiderivative size = 139 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2} d^6} \]

[Out]

-1/24*(c*x^2+b*x+a)^(3/2)/c^2/d^6/(2*c*x+b)^3-1/10*(c*x^2+b*x+a)^(5/2)/c/d^6/(2*c*x+b)^5+1/64*arctanh(1/2*(2*c
*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)/d^6-1/32*(c*x^2+b*x+a)^(1/2)/c^3/d^6/(2*c*x+b)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {698, 635, 212} \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2} d^6}-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5} \]

[In]

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

[Out]

-1/32*Sqrt[a + b*x + c*x^2]/(c^3*d^6*(b + 2*c*x)) - (a + b*x + c*x^2)^(3/2)/(24*c^2*d^6*(b + 2*c*x)^3) - (a +
b*x + c*x^2)^(5/2)/(10*c*d^6*(b + 2*c*x)^5) + ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(64*c^(7/
2)*d^6)

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 698

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 1))), x] - Dist[b*(p/(d*e*(m + 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] &&
GtQ[p, 0] && LtQ[m, -1] &&  !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0]) && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^4} \, dx}{4 c d^2} \\ & = -\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^2} \, dx}{16 c^2 d^4} \\ & = -\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{64 c^3 d^6} \\ & = -\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\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 )}{32 c^3 d^6} \\ & = -\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2} d^6} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=-\frac {\left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{2},-\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{320 c^3 d^6 (b+2 c x)^5 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]

[In]

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

[Out]

-1/320*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -5/2, -3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]
)/(c^3*d^6*(b + 2*c*x)^5*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(439\) vs. \(2(117)=234\).

Time = 4.46 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.17

method	result	size
default	\(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{5 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{5}}+\frac {8 c^{2} \left (-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3}}+\frac {16 c^{2} \left (-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {24 c^{2} \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{6}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{4}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{2}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\sqrt {c}\, \left (x +\frac {b}{2 c}\right )+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}\right )}{5 \left (4 a c -b^{2}\right )}}{64 d^{6} c^{6}}\)	\(440\)

[In]

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

[Out]

1/64/d^6/c^6*(-4/5/(4*a*c-b^2)*c/(x+1/2/c*b)^5*((x+1/2/c*b)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+8/5*c^2/(4*a*c-b^2)*(
-4/3/(4*a*c-b^2)*c/(x+1/2/c*b)^3*((x+1/2/c*b)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+16/3*c^2/(4*a*c-b^2)*(-4/(4*a*c-b^2
)*c/(x+1/2/c*b)*((x+1/2/c*b)^2*c+1/4*(4*a*c-b^2)/c)^(7/2)+24*c^2/(4*a*c-b^2)*(1/6*(x+1/2/c*b)*((x+1/2/c*b)^2*c
+1/4*(4*a*c-b^2)/c)^(5/2)+5/24*(4*a*c-b^2)/c*(1/4*(x+1/2/c*b)*((x+1/2/c*b)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)+3/16*(
4*a*c-b^2)/c*(1/2*(x+1/2/c*b)*((x+1/2/c*b)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln(c^(1/2)*(x+
1/2/c*b)+((x+1/2/c*b)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))))))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (117) = 234\).

Time = 1.51 (sec) , antiderivative size = 549, normalized size of antiderivative = 3.95 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=\left [\frac {15 \, {\left (32 \, c^{5} x^{5} + 80 \, b c^{4} x^{4} + 80 \, b^{2} c^{3} x^{3} + 40 \, b^{3} c^{2} x^{2} + 10 \, b^{4} c x + b^{5}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (368 \, c^{5} x^{4} + 736 \, b c^{4} x^{3} + 15 \, b^{4} c + 20 \, a b^{2} c^{2} + 48 \, a^{2} c^{3} + 4 \, {\left (127 \, b^{2} c^{3} + 44 \, a c^{4}\right )} x^{2} + 4 \, {\left (35 \, b^{3} c^{2} + 44 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1920 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}}, -\frac {15 \, {\left (32 \, c^{5} x^{5} + 80 \, b c^{4} x^{4} + 80 \, b^{2} c^{3} x^{3} + 40 \, b^{3} c^{2} x^{2} + 10 \, b^{4} c x + b^{5}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (368 \, c^{5} x^{4} + 736 \, b c^{4} x^{3} + 15 \, b^{4} c + 20 \, a b^{2} c^{2} + 48 \, a^{2} c^{3} + 4 \, {\left (127 \, b^{2} c^{3} + 44 \, a c^{4}\right )} x^{2} + 4 \, {\left (35 \, b^{3} c^{2} + 44 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{960 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}}\right ] \]

[In]

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

[Out]

[1/1920*(15*(32*c^5*x^5 + 80*b*c^4*x^4 + 80*b^2*c^3*x^3 + 40*b^3*c^2*x^2 + 10*b^4*c*x + b^5)*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) - 4*(368*c^5*x^4 + 736*b*c^4*x^3
+ 15*b^4*c + 20*a*b^2*c^2 + 48*a^2*c^3 + 4*(127*b^2*c^3 + 44*a*c^4)*x^2 + 4*(35*b^3*c^2 + 44*a*b*c^3)*x)*sqrt(
c*x^2 + b*x + a))/(32*c^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 + 10*b^4*c^5*d^
6*x + b^5*c^4*d^6), -1/960*(15*(32*c^5*x^5 + 80*b*c^4*x^4 + 80*b^2*c^3*x^3 + 40*b^3*c^2*x^2 + 10*b^4*c*x + b^5
)*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)) + 2*(368*c^5*x^4 + 7
36*b*c^4*x^3 + 15*b^4*c + 20*a*b^2*c^2 + 48*a^2*c^3 + 4*(127*b^2*c^3 + 44*a*c^4)*x^2 + 4*(35*b^3*c^2 + 44*a*b*
c^3)*x)*sqrt(c*x^2 + b*x + a))/(32*c^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 +
10*b^4*c^5*d^6*x + b^5*c^4*d^6)]

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx}{d^{6}} \]

[In]

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

[Out]

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*
c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*
x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integr
al(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c*
*4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x +
60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(2
*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*
x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x +
60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x))/d**6

Maxima [F(-2)]

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

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,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. 1013 vs. \(2 (117) = 234\).

Time = 0.58 (sec) , antiderivative size = 1013, normalized size of antiderivative = 7.29 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=-\frac {\log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{64 \, c^{\frac {7}{2}} d^{6}} - \frac {720 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} b^{2} c^{4} - 2880 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} a c^{5} + 2880 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} b^{3} c^{\frac {7}{2}} - 11520 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} a b c^{\frac {9}{2}} + 5400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} b^{4} c^{3} - 23040 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} a b^{2} c^{4} + 5760 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} a^{2} c^{5} + 6120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} b^{5} c^{\frac {5}{2}} - 28800 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a b^{3} c^{\frac {7}{2}} + 17280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a^{2} b c^{\frac {9}{2}} + 4640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} b^{6} c^{2} - 25080 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a b^{4} c^{3} + 28320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{2} b^{2} c^{4} - 8960 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{3} c^{5} + 2440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{7} c^{\frac {3}{2}} - 15600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b^{5} c^{\frac {5}{2}} + 27840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} b^{3} c^{\frac {7}{2}} - 17920 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{3} b c^{\frac {9}{2}} + 880 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{8} c - 6760 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{6} c^{2} + 17160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} b^{4} c^{3} - 17920 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{3} b^{2} c^{4} + 4480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{4} c^{5} + 200 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{9} \sqrt {c} - 1840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{7} c^{\frac {3}{2}} + 6120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b^{5} c^{\frac {5}{2}} - 8960 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{3} b^{3} c^{\frac {7}{2}} + 4480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{4} b c^{\frac {9}{2}} + 23 \, b^{10} - 260 \, a b^{8} c + 1160 \, a^{2} b^{6} c^{2} - 2600 \, a^{3} b^{4} c^{3} + 2960 \, a^{4} b^{2} c^{4} - 1472 \, a^{5} c^{5}}{960 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{5} c^{\frac {7}{2}} d^{6}} \]

[In]

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

[Out]

-1/64*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/(c^(7/2)*d^6) - 1/960*(720*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^8*b^2*c^4 - 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^5 + 2880*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^7*b^3*c^(7/2) - 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b*c^(9/2) + 5400*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^6*b^4*c^3 - 23040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^4 + 5760*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^6*a^2*c^5 + 6120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^5*c^(5/2) - 28800*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^5*a*b^3*c^(7/2) + 17280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b*c^(9/2) + 4640*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^4*b^6*c^2 - 25080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^4*c^3 + 28320*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^4*a^2*b^2*c^4 - 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*c^5 + 2440*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^3*b^7*c^(3/2) - 15600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^5*c^(5/2) + 27840*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^3*c^(7/2) - 17920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b*c^(9/
2) + 880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^8*c - 6760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^6*c^2 +
17160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^4*c^3 - 17920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^2*
c^4 + 4480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*c^5 + 200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^9*sqrt(c)
- 1840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^7*c^(3/2) + 6120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^5*c
^(5/2) - 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^3*c^(7/2) + 4480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a
^4*b*c^(9/2) + 23*b^10 - 260*a*b^8*c + 1160*a^2*b^6*c^2 - 2600*a^3*b^4*c^3 + 2960*a^4*b^2*c^4 - 1472*a^5*c^5)/
((2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^5
*c^(7/2)*d^6)

Mupad [F(-1)]

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

[In]

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

[Out]

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