Integrand size = 26, antiderivative size = 147 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^5} \, dx=\frac {15 \sqrt {a+b x+c x^2}}{256 c^3 d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{64 c^2 d^5 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}-\frac {15 \sqrt {b^2-4 a c} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{512 c^{7/2} d^5} \] Output:
15/256*(c*x^2+b*x+a)^(1/2)/c^3/d^5-5/64*(c*x^2+b*x+a)^(3/2)/c^2/d^5/(2*c*x +b)^2-1/8*(c*x^2+b*x+a)^(5/2)/c/d^5/(2*c*x+b)^4-15/512*(-4*a*c+b^2)^(1/2)* arctan(2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^(7/2)/d^5
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^5} \, dx=\frac {2 (a+x (b+c x))^{7/2} \operatorname {Hypergeometric2F1}\left (3,\frac {7}{2},\frac {9}{2},\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{7 \left (b^2-4 a c\right )^3 d^5} \] Input:
Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^5,x]
Output:
(2*(a + x*(b + c*x))^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(7*(b^2 - 4*a*c)^3*d^5)
Time = 0.31 (sec) , antiderivative size = 157, 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, 1109, 1112, 218}
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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^5} \, dx\) |
| \(\Big \downarrow \) 1108 |
| \(\displaystyle \frac {5 \int \frac {\left (c x^2+b x+a\right )^{3/2}}{d^3 (b+2 c x)^3}dx}{16 c d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \int \frac {\left (c x^2+b x+a\right )^{3/2}}{(b+2 c x)^3}dx}{16 c d^5}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}\) |
| \(\Big \downarrow \) 1108 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {\sqrt {c x^2+b x+a}}{b+2 c x}dx}{8 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{4 c (b+2 c x)^2}\right )}{16 c d^5}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}\) |
| \(\Big \downarrow \) 1109 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {\sqrt {a+b x+c x^2}}{2 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(b+2 c x) \sqrt {c x^2+b x+a}}dx}{4 c}\right )}{8 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{4 c (b+2 c x)^2}\right )}{16 c d^5}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}\) |
| \(\Big \downarrow \) 1112 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {\sqrt {a+b x+c x^2}}{2 c}-\left (b^2-4 a c\right ) \int \frac {1}{8 \left (c x^2+b x+a\right ) c^2+2 \left (b^2-4 a c\right ) c}d\sqrt {c x^2+b x+a}\right )}{8 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{4 c (b+2 c x)^2}\right )}{16 c d^5}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {\sqrt {a+b x+c x^2}}{2 c}-\frac {\sqrt {b^2-4 a c} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{4 c^{3/2}}\right )}{8 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{4 c (b+2 c x)^2}\right )}{16 c d^5}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}\) |
Input:
Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^5,x]
Output:
-1/8*(a + b*x + c*x^2)^(5/2)/(c*d^5*(b + 2*c*x)^4) + (5*(-1/4*(a + b*x + c *x^2)^(3/2)/(c*(b + 2*c*x)^2) + (3*(Sqrt[a + b*x + c*x^2]/(2*c) - (Sqrt[b^ 2 - 4*a*c]*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(4 *c^(3/2))))/(8*c)))/(16*c*d^5)
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[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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]
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 + 2*p + 1))), x ] - Simp[d*p*((b^2 - 4*a*c)/(b*e*(m + 2*p + 1))) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b* e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[p, 0] && !LtQ[m, -1] && !(IGtQ[(m - 1 )/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p])) && RationalQ[m] && IntegerQ[2*p]
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symb ol] :> Simp[4*c Subst[Int[1/(b^2*e - 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Time = 1.64 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.19
| method | result | size |
| pseudoelliptic | \(-\frac {\frac {15 \left (2 c x +b \right )^{4} \left (a c -\frac {b^{2}}{4}\right ) \operatorname {arctanh}\left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, c}{\sqrt {4 a \,c^{2}-b^{2} c}}\right )}{16}+\left (-4 c^{4} x^{4}+\left (-8 b \,x^{3}+\frac {9}{2} a \,x^{2}\right ) c^{3}+\left (-\frac {57}{8} b^{2} x^{2}+\frac {9}{2} a b x +a^{2}\right ) c^{2}+\frac {5 b^{2} \left (-5 b x +a \right ) c}{8}-\frac {15 b^{4}}{32}\right ) \sqrt {c \,x^{2}+b x +a}\, \sqrt {4 a \,c^{2}-b^{2} c}}{8 \sqrt {4 a \,c^{2}-b^{2} c}\, \left (2 c x +b \right )^{4} c^{3} d^{5}}\) | \(175\) |
| default | \(\frac {-\frac {c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}+\frac {3 c^{2} \left (-\frac {2 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {10 c^{2} \left (\frac {\left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{5}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 c}\right )}{4 c}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}}{32 d^{5} c^{5}}\) | \(391\) |
| risch | \(\frac {\sqrt {c \,x^{2}+b x +a}}{32 c^{3} d^{5}}+\frac {-\frac {\left (12 a c -3 b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{c \sqrt {\frac {4 a c -b^{2}}{c}}}+\frac {\left (48 a^{2} c^{2}-24 c a \,b^{2}+3 b^{4}\right ) \left (-\frac {2 c \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {4 c^{2} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{8 c^{3}}+\frac {\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (-\frac {c \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}-\frac {3 c^{2} \left (-\frac {2 c \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {4 c^{2} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 a c -b^{2}}\right )}{32 c^{5}}}{64 c^{3} d^{5}}\) | \(604\) |
Input:
int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^5,x,method=_RETURNVERBOSE)
Output:
-1/8/(4*a*c^2-b^2*c)^(1/2)*(15/16*(2*c*x+b)^4*(a*c-1/4*b^2)*arctanh(2*(c*x ^2+b*x+a)^(1/2)*c/(4*a*c^2-b^2*c)^(1/2))+(-4*c^4*x^4+(-8*b*x^3+9/2*a*x^2)* c^3+(-57/8*b^2*x^2+9/2*a*b*x+a^2)*c^2+5/8*b^2*(-5*b*x+a)*c-15/32*b^4)*(c*x ^2+b*x+a)^(1/2)*(4*a*c^2-b^2*c)^(1/2))/(2*c*x+b)^4/c^3/d^5
Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (123) = 246\).
Time = 0.75 (sec) , antiderivative size = 535, normalized size of antiderivative = 3.64 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^5} \, dx=\left [\frac {15 \, {\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt {-\frac {b^{2} - 4 \, a c}{c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {c x^{2} + b x + a} c \sqrt {-\frac {b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, {\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} + 15 \, b^{4} - 20 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \, {\left (19 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{2} + 4 \, {\left (25 \, b^{3} c - 36 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1024 \, {\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}}, \frac {15 \, {\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c}} \arctan \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} c \sqrt {\frac {b^{2} - 4 \, a c}{c}}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} + 15 \, b^{4} - 20 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \, {\left (19 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{2} + 4 \, {\left (25 \, b^{3} c - 36 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{512 \, {\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}}\right ] \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^5,x, algorithm="fricas")
Output:
[1/1024*(15*(16*c^4*x^4 + 32*b*c^3*x^3 + 24*b^2*c^2*x^2 + 8*b^3*c*x + b^4) *sqrt(-(b^2 - 4*a*c)/c)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(c *x^2 + b*x + a)*c*sqrt(-(b^2 - 4*a*c)/c))/(4*c^2*x^2 + 4*b*c*x + b^2)) + 4 *(128*c^4*x^4 + 256*b*c^3*x^3 + 15*b^4 - 20*a*b^2*c - 32*a^2*c^2 + 12*(19* b^2*c^2 - 12*a*c^3)*x^2 + 4*(25*b^3*c - 36*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a))/(16*c^7*d^5*x^4 + 32*b*c^6*d^5*x^3 + 24*b^2*c^5*d^5*x^2 + 8*b^3*c^4*d^ 5*x + b^4*c^3*d^5), 1/512*(15*(16*c^4*x^4 + 32*b*c^3*x^3 + 24*b^2*c^2*x^2 + 8*b^3*c*x + b^4)*sqrt((b^2 - 4*a*c)/c)*arctan(-2*sqrt(c*x^2 + b*x + a)*c *sqrt((b^2 - 4*a*c)/c)/(b^2 - 4*a*c)) + 2*(128*c^4*x^4 + 256*b*c^3*x^3 + 1 5*b^4 - 20*a*b^2*c - 32*a^2*c^2 + 12*(19*b^2*c^2 - 12*a*c^3)*x^2 + 4*(25*b ^3*c - 36*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a))/(16*c^7*d^5*x^4 + 32*b*c^6*d^ 5*x^3 + 24*b^2*c^5*d^5*x^2 + 8*b^3*c^4*d^5*x + b^4*c^3*d^5)]
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^5} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx}{d^{5}} \] Input:
integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**5,x)
Output:
(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**5 + 10*b**4*c*x + 40*b**3*c**2*x **2 + 80*b**2*c**3*x**3 + 80*b*c**4*x**4 + 32*c**5*x**5), x) + Integral(b* *2*x**2*sqrt(a + b*x + c*x**2)/(b**5 + 10*b**4*c*x + 40*b**3*c**2*x**2 + 8 0*b**2*c**3*x**3 + 80*b*c**4*x**4 + 32*c**5*x**5), x) + Integral(c**2*x**4 *sqrt(a + b*x + c*x**2)/(b**5 + 10*b**4*c*x + 40*b**3*c**2*x**2 + 80*b**2* c**3*x**3 + 80*b*c**4*x**4 + 32*c**5*x**5), x) + Integral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**5 + 10*b**4*c*x + 40*b**3*c**2*x**2 + 80*b**2*c**3*x**3 + 80*b*c**4*x**4 + 32*c**5*x**5), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**5 + 10*b**4*c*x + 40*b**3*c**2*x**2 + 80*b**2*c**3*x**3 + 80* b*c**4*x**4 + 32*c**5*x**5), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x** 2)/(b**5 + 10*b**4*c*x + 40*b**3*c**2*x**2 + 80*b**2*c**3*x**3 + 80*b*c**4 *x**4 + 32*c**5*x**5), x))/d**5
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^5} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^5,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. 692 vs. \(2 (123) = 246\).
Time = 0.79 (sec) , antiderivative size = 692, normalized size of antiderivative = 4.71 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^5} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^5,x, algorithm="giac")
Output:
1/1024*d^2*(2*sqrt(-b^2*c*d^2/(2*c*d*x + b*d)^2 + 4*a*c^2*d^2/(2*c*d*x + b *d)^2 + c)*(9*(b^6*c^7*d^10*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) - 12*a*b^ 4*c^8*d^10*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) + 48*a^2*b^2*c^9*d^10*sgn( 1/(2*c*d*x + b*d))*sgn(c)*sgn(d) - 64*a^3*c^10*d^10*sgn(1/(2*c*d*x + b*d)) *sgn(c)*sgn(d))/(b^4*c^10*d^15*abs(c) - 8*a*b^2*c^11*d^15*abs(c) + 16*a^2* c^12*d^15*abs(c)) - 2*(b^8*c^9*d^14*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) - 16*a*b^6*c^10*d^14*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) + 96*a^2*b^4*c^11 *d^14*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d) - 256*a^3*b^2*c^12*d^14*sgn(1/( 2*c*d*x + b*d))*sgn(c)*sgn(d) + 256*a^4*c^13*d^14*sgn(1/(2*c*d*x + b*d))*s gn(c)*sgn(d))/((b^4*c^10*d^15*abs(c) - 8*a*b^2*c^11*d^15*abs(c) + 16*a^2*c ^12*d^15*abs(c))*(2*c*d*x + b*d)^2*c^2*d^2))/((2*c*d*x + b*d)*c*d) - 15*sq rt(-b^2*c + 4*a*c^2)*log((sqrt(-b^2*c*d^2/(2*c*d*x + b*d)^2 + 4*a*c^2*d^2/ (2*c*d*x + b*d)^2 + c) + sqrt(-b^2*c^3*d^4 + 4*a*c^4*d^4)/((2*c*d*x + b*d) *c*d))^2)*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d)/(c^5*d^7) + 32*sqrt(-b^2*c + 4*a*c^2)*sgn(1/(2*c*d*x + b*d))*sgn(c)*sgn(d)/(((sqrt(-b^2*c*d^2/(2*c*d* x + b*d)^2 + 4*a*c^2*d^2/(2*c*d*x + b*d)^2 + c) + sqrt(-b^2*c^3*d^4 + 4*a* c^4*d^4)/((2*c*d*x + b*d)*c*d))^2 - c)*c^4*d^7))*abs(c)
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^5} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^5} \,d x \] Input:
int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^5,x)
Output:
int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^5, x)
Time = 1.13 (sec) , antiderivative size = 916, normalized size of antiderivative = 6.23 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^5} \, dx =\text {Too large to display} \] Input:
int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^5,x)
Output:
(15*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt (a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**4 + 120*sqrt(c)*sqr t(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x* *2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c*x + 360*sqrt(c)*sqrt(4*a*c - b **2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2 *c*x)/sqrt(4*a*c - b**2))*b**2*c**2*x**2 + 480*sqrt(c)*sqrt(4*a*c - b**2)* log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x) /sqrt(4*a*c - b**2))*b*c**3*x**3 + 240*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4* a*c - b**2))*c**4*x**4 - 15*sqrt(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b **2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b **4 - 120*sqrt(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*s qrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c*x - 360*sqrt (c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c**2*x**2 - 480*sqrt(c)*sqrt (4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**3*x**3 - 240*sqrt(c)*sqrt(4*a*c - b* *2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x )/sqrt(4*a*c - b**2))*c**4*x**4 - 64*sqrt(a + b*x + c*x**2)*a**2*c**3 - 40 *sqrt(a + b*x + c*x**2)*a*b**2*c**2 - 288*sqrt(a + b*x + c*x**2)*a*b*c*...