Integrand size = 26, antiderivative size = 145 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\frac {5 (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3 d^4}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}-\frac {5 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2} d^4} \] Output:
5/64*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^3/d^4-5/24*(c*x^2+b*x+a)^(3/2)/c^2/d^ 4/(2*c*x+b)-1/6*(c*x^2+b*x+a)^(5/2)/c/d^4/(2*c*x+b)^3-5/128*(-4*a*c+b^2)*a rctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)/d^4
Leaf count is larger than twice the leaf count of optimal. \(331\) vs. \(2(145)=290\).
Time = 2.90 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.28 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\frac {\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (15 b^4+80 b^3 c x+32 b c^2 x \left (-7 a+3 c x^2\right )+8 b^2 c \left (-5 a+16 c x^2\right )+16 c^2 \left (-2 a^2-14 a c x^2+3 c^2 x^4\right )\right )}{(b+2 c x)^3}+\frac {12 \sqrt {b^2-4 a c} \left (b^4-2 a b^2 c-8 a^2 c^2\right ) \arctan \left (\frac {\sqrt {c} \sqrt {b^2-4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{b^3}-15 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )+\frac {12 \sqrt {-b^2+4 a c} \left (b^4-2 a b^2 c-8 a^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {-b^2+4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{b^3}}{192 c^{7/2} d^4} \] Input:
Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^4,x]
Output:
((Sqrt[c]*Sqrt[a + x*(b + c*x)]*(15*b^4 + 80*b^3*c*x + 32*b*c^2*x*(-7*a + 3*c*x^2) + 8*b^2*c*(-5*a + 16*c*x^2) + 16*c^2*(-2*a^2 - 14*a*c*x^2 + 3*c^2 *x^4)))/(b + 2*c*x)^3 + (12*Sqrt[b^2 - 4*a*c]*(b^4 - 2*a*b^2*c - 8*a^2*c^2 )*ArcTan[(Sqrt[c]*Sqrt[b^2 - 4*a*c]*x)/(Sqrt[a]*(b + 2*c*x) - b*Sqrt[a + x *(b + c*x)])])/b^3 - 15*(b^2 - 4*a*c)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt [a + x*(b + c*x)])] + (12*Sqrt[-b^2 + 4*a*c]*(b^4 - 2*a*b^2*c - 8*a^2*c^2) *ArcTanh[(Sqrt[c]*Sqrt[-b^2 + 4*a*c]*x)/(Sqrt[a]*(b + 2*c*x) - b*Sqrt[a + x*(b + c*x)])])/b^3)/(192*c^(7/2)*d^4)
Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1108, 27, 1108, 1087, 1092, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx\) |
| \(\Big \downarrow \) 1108 |
| \(\displaystyle \frac {5 \int \frac {\left (c x^2+b x+a\right )^{3/2}}{d^2 (b+2 c x)^2}dx}{12 c d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \int \frac {\left (c x^2+b x+a\right )^{3/2}}{(b+2 c x)^2}dx}{12 c d^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}\) |
| \(\Big \downarrow \) 1108 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \sqrt {c x^2+b x+a}dx}{4 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 c (b+2 c x)}\right )}{12 c d^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{4 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 c (b+2 c x)}\right )}{12 c d^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{4 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 c (b+2 c x)}\right )}{12 c d^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{4 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 c (b+2 c x)}\right )}{12 c d^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}\) |
Input:
Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^4,x]
Output:
-1/6*(a + b*x + c*x^2)^(5/2)/(c*d^4*(b + 2*c*x)^3) + (5*(-1/2*(a + b*x + c *x^2)^(3/2)/(c*(b + 2*c*x)) + (3*(((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c ) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]) /(8*c^(3/2))))/(4*c)))/(12*c*d^4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[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] && 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]
Leaf count of result is larger than twice the leaf count of optimal. \(328\) vs. \(2(123)=246\).
Time = 1.48 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.27
| method | result | size |
| risch | \(\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{64 c^{3} d^{4}}+\frac {-\frac {\left (96 a^{2} c^{2}-48 c a \,b^{2}+6 b^{4}\right ) \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}{c \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {\left (128 a^{3} c^{3}-96 a^{2} b^{2} c^{2}+24 a \,b^{4} c -2 b^{6}\right ) \left (-\frac {4 c \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3}}+\frac {32 c^{3} \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}{3 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )}\right )}{16 c^{4}}-\frac {5 b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+20 a \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{3} d^{4}}\) | \(329\) |
| default | \(\frac {-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3}}+\frac {16 c^{2} \left (-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {24 c^{2} \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{6}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{4}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}{2}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\sqrt {c}\, \left (x +\frac {b}{2 c}\right )+\sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}}{16 d^{4} c^{4}}\) | \(367\) |
Input:
int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^4,x,method=_RETURNVERBOSE)
Output:
1/64*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^3/d^4+1/128/c^3*(-(96*a^2*c^2-48*a*b^ 2*c+6*b^4)/c/(4*a*c-b^2)/(x+1/2*b/c)*(c*(x+1/2*b/c)^2+1/4*(4*a*c-b^2)/c)^( 1/2)+1/16*(128*a^3*c^3-96*a^2*b^2*c^2+24*a*b^4*c-2*b^6)/c^4*(-4/3/(4*a*c-b ^2)*c/(x+1/2*b/c)^3*(c*(x+1/2*b/c)^2+1/4*(4*a*c-b^2)/c)^(1/2)+32/3*c^3/(4* a*c-b^2)^2/(x+1/2*b/c)*(c*(x+1/2*b/c)^2+1/4*(4*a*c-b^2)/c)^(1/2))-5*b^2*ln ((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+20*a*c^(1/2)*ln((1/2*b+c *x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))/d^4
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (123) = 246\).
Time = 0.47 (sec) , antiderivative size = 529, normalized size of antiderivative = 3.65 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\left [-\frac {15 \, {\left (b^{5} - 4 \, a b^{3} c + 8 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 12 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 6 \, {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (48 \, c^{5} x^{4} + 96 \, b c^{4} x^{3} + 15 \, b^{4} c - 40 \, a b^{2} c^{2} - 32 \, a^{2} c^{3} + 32 \, {\left (4 \, b^{2} c^{3} - 7 \, a c^{4}\right )} x^{2} + 16 \, {\left (5 \, b^{3} c^{2} - 14 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, {\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\right )}}, \frac {15 \, {\left (b^{5} - 4 \, a b^{3} c + 8 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 12 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 6 \, {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (48 \, c^{5} x^{4} + 96 \, b c^{4} x^{3} + 15 \, b^{4} c - 40 \, a b^{2} c^{2} - 32 \, a^{2} c^{3} + 32 \, {\left (4 \, b^{2} c^{3} - 7 \, a c^{4}\right )} x^{2} + 16 \, {\left (5 \, b^{3} c^{2} - 14 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, {\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\right )}}\right ] \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^4,x, algorithm="fricas")
Output:
[-1/768*(15*(b^5 - 4*a*b^3*c + 8*(b^2*c^3 - 4*a*c^4)*x^3 + 12*(b^3*c^2 - 4 *a*b*c^3)*x^2 + 6*(b^4*c - 4*a*b^2*c^2)*x)*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*(48*c^5 *x^4 + 96*b*c^4*x^3 + 15*b^4*c - 40*a*b^2*c^2 - 32*a^2*c^3 + 32*(4*b^2*c^3 - 7*a*c^4)*x^2 + 16*(5*b^3*c^2 - 14*a*b*c^3)*x)*sqrt(c*x^2 + b*x + a))/(8 *c^7*d^4*x^3 + 12*b*c^6*d^4*x^2 + 6*b^2*c^5*d^4*x + b^3*c^4*d^4), 1/384*(1 5*(b^5 - 4*a*b^3*c + 8*(b^2*c^3 - 4*a*c^4)*x^3 + 12*(b^3*c^2 - 4*a*b*c^3)* x^2 + 6*(b^4*c - 4*a*b^2*c^2)*x)*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*(48*c^5*x^4 + 96*b*c^4* x^3 + 15*b^4*c - 40*a*b^2*c^2 - 32*a^2*c^3 + 32*(4*b^2*c^3 - 7*a*c^4)*x^2 + 16*(5*b^3*c^2 - 14*a*b*c^3)*x)*sqrt(c*x^2 + b*x + a))/(8*c^7*d^4*x^3 + 1 2*b*c^6*d^4*x^2 + 6*b^2*c^5*d^4*x + b^3*c^4*d^4)]
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx}{d^{4}} \] Input:
integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**4,x)
Output:
(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x* *2 + 32*b*c**3*x**3 + 16*c**4*x**4), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**3 + 16*c** 4*x**4), x) + Integral(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**3 + 16*c**4*x**4), x) + Integral(2*a*b *x*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c* *3*x**3 + 16*c**4*x**4), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x**2)/( b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**3 + 16*c**4*x**4), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2 *c**2*x**2 + 32*b*c**3*x**3 + 16*c**4*x**4), x))/d**4
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^4,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (123) = 246\).
Time = 0.49 (sec) , antiderivative size = 612, normalized size of antiderivative = 4.22 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\frac {1}{64} \, \sqrt {c x^{2} + b x + a} {\left (\frac {2 \, x}{c^{2} d^{4}} + \frac {b}{c^{3} d^{4}}\right )} + \frac {5 \, {\left (b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {7}{2}} d^{4}} + \frac {36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} b^{4} c^{2} - 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a b^{2} c^{3} + 576 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{2} c^{4} + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{5} c^{\frac {3}{2}} - 576 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b^{3} c^{\frac {5}{2}} + 1152 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} b c^{\frac {7}{2}} + 66 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{6} c - 576 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{4} c^{2} + 1440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} b^{2} c^{3} - 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{3} c^{4} + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{7} \sqrt {c} - 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{5} c^{\frac {3}{2}} + 864 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b^{3} c^{\frac {5}{2}} - 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{3} b c^{\frac {7}{2}} + 7 \, b^{8} - 82 \, a b^{6} c + 348 \, a^{2} b^{4} c^{2} - 640 \, a^{3} b^{2} c^{3} + 448 \, a^{4} c^{4}}{192 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{3} c^{\frac {7}{2}} d^{4}} \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^4,x, algorithm="giac")
Output:
1/64*sqrt(c*x^2 + b*x + a)*(2*x/(c^2*d^4) + b/(c^3*d^4)) + 5/128*(b^2 - 4* a*c)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/(c^(7/2)* d^4) + 1/192*(36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*c^2 - 288*(sqrt (c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^3 + 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^4 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*c^(3/2 ) - 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*c^(5/2) + 1152*(sqrt(c )*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c^(7/2) + 66*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^6*c - 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*c^2 + 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c^3 - 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^4 + 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) )*b^7*sqrt(c) - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*c^(3/2) + 86 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*c^(5/2) - 768*(sqrt(c)*x - s qrt(c*x^2 + b*x + a))*a^3*b*c^(7/2) + 7*b^8 - 82*a*b^6*c + 348*a^2*b^4*c^2 - 640*a^3*b^2*c^3 + 448*a^4*c^4)/((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)^3*c^( 7/2)*d^4)
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^4} \,d x \] Input:
int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^4,x)
Output:
int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^4, x)
Time = 0.24 (sec) , antiderivative size = 675, normalized size of antiderivative = 4.66 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\frac {192 \sqrt {c \,x^{2}+b x +a}\, c^{5} x^{4}-30 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{5}+384 \sqrt {c \,x^{2}+b x +a}\, b \,c^{4} x^{3}-30 \sqrt {c}\, b^{4} c x -60 \sqrt {c}\, b^{3} c^{2} x^{2}-40 \sqrt {c}\, b^{2} c^{3} x^{3}-128 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{3}+60 \sqrt {c \,x^{2}+b x +a}\, b^{4} c +1440 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a b \,c^{3} x^{2}+120 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{3} c -180 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{4} c x +120 \sqrt {c}\, a \,b^{2} c^{2} x +20 \sqrt {c}\, a \,b^{3} c -896 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{3} x -160 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{2}-896 \sqrt {c \,x^{2}+b x +a}\, a \,c^{4} x^{2}+320 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{2} x -5 \sqrt {c}\, b^{5}+960 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{4} x^{3}-360 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c^{2} x^{2}-240 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c^{3} x^{3}+240 \sqrt {c}\, a b \,c^{3} x^{2}+160 \sqrt {c}\, a \,c^{4} x^{3}+720 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c^{2} x +512 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{3} x^{2}}{768 c^{4} d^{4} \left (8 c^{3} x^{3}+12 b \,c^{2} x^{2}+6 b^{2} c x +b^{3}\right )} \] Input:
int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^4,x)
Output:
( - 128*sqrt(a + b*x + c*x**2)*a**2*c**3 - 160*sqrt(a + b*x + c*x**2)*a*b* *2*c**2 - 896*sqrt(a + b*x + c*x**2)*a*b*c**3*x - 896*sqrt(a + b*x + c*x** 2)*a*c**4*x**2 + 60*sqrt(a + b*x + c*x**2)*b**4*c + 320*sqrt(a + b*x + c*x **2)*b**3*c**2*x + 512*sqrt(a + b*x + c*x**2)*b**2*c**3*x**2 + 384*sqrt(a + b*x + c*x**2)*b*c**4*x**3 + 192*sqrt(a + b*x + c*x**2)*c**5*x**4 + 120*s qrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b** 2))*a*b**3*c + 720*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c *x)/sqrt(4*a*c - b**2))*a*b**2*c**2*x + 1440*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c**3*x**2 + 960*sqrt (c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2)) *a*c**4*x**3 - 30*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c* x)/sqrt(4*a*c - b**2))*b**5 - 180*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c* x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**4*c*x - 360*sqrt(c)*log((2*sqrt( c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c**2*x**2 - 240*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a* c - b**2))*b**2*c**3*x**3 + 20*sqrt(c)*a*b**3*c + 120*sqrt(c)*a*b**2*c**2* x + 240*sqrt(c)*a*b*c**3*x**2 + 160*sqrt(c)*a*c**4*x**3 - 5*sqrt(c)*b**5 - 30*sqrt(c)*b**4*c*x - 60*sqrt(c)*b**3*c**2*x**2 - 40*sqrt(c)*b**2*c**3*x* *3)/(768*c**4*d**4*(b**3 + 6*b**2*c*x + 12*b*c**2*x**2 + 8*c**3*x**3))