Integrand size = 22, antiderivative size = 118 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {16 \left (c d^2-b d e+a e^2\right ) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \] Output:
-2/3*(e*x+d)^2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2) +16/3*(a*e^2-b*d*e+c*d^2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)^2/(c*x^2 +b*x+a)^(1/2)
Time = 1.24 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.61 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (-6 b^2 (a e-c d x) \left (d^2-6 d e x+e^2 x^2\right )+b^3 \left (-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )+12 b (d-e x) \left (2 a^2 e^2+2 c^2 d^2 x^2+a c (d-e x)^2\right )-8 \left (2 a^3 e^3-2 c^3 d^3 x^3+3 a^2 c e \left (d^2+e^2 x^2\right )-3 a c^2 d x \left (d^2+e^2 x^2\right )\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \] Input:
Integrate[(d + e*x)^3/(a + b*x + c*x^2)^(5/2),x]
Output:
(2*(-6*b^2*(a*e - c*d*x)*(d^2 - 6*d*e*x + e^2*x^2) + b^3*(-d^3 - 9*d^2*e*x + 9*d*e^2*x^2 + e^3*x^3) + 12*b*(d - e*x)*(2*a^2*e^2 + 2*c^2*d^2*x^2 + a* c*(d - e*x)^2) - 8*(2*a^3*e^3 - 2*c^3*d^3*x^3 + 3*a^2*c*e*(d^2 + e^2*x^2) - 3*a*c^2*d*x*(d^2 + e^2*x^2))))/(3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2 ))
Time = 0.24 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1153, 1158}
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.
\(\displaystyle \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1153 |
\(\displaystyle -\frac {8 \left (a e^2-b d e+c d^2\right ) \int \frac {d+e x}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1158 |
\(\displaystyle \frac {16 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
Input:
Int[(d + e*x)^3/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*(d + e*x)^2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (16*(c*d^2 - b*d*e + a*e^2)*(b*d - 2*a*e + (2*c*d - b*e )*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*(2*p + 3)*((c*d^2 - b*d*e + a*e^2)/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && LtQ[p, -1]
Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(286\) vs. \(2(110)=220\).
Time = 1.11 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.43
method | result | size |
trager | \(-\frac {2 \left (12 a b c \,e^{3} x^{3}-24 a \,c^{2} d \,e^{2} x^{3}-e^{3} x^{3} b^{3}-6 b^{2} c d \,e^{2} x^{3}+24 b \,c^{2} d^{2} e \,x^{3}-16 d^{3} c^{3} x^{3}+24 a^{2} c \,e^{3} x^{2}+6 x^{2} a \,b^{2} e^{3}-36 a b c d \,e^{2} x^{2}-9 x^{2} b^{3} d \,e^{2}+36 b^{2} c \,d^{2} e \,x^{2}-24 b \,c^{2} d^{3} x^{2}+24 a^{2} b \,e^{3} x -36 x a \,b^{2} d \,e^{2}+36 a b c \,d^{2} e x -24 a \,c^{2} d^{3} x +9 b^{3} d^{2} e x -6 b^{2} c \,d^{3} x +16 e^{3} a^{3}-24 a^{2} b d \,e^{2}+24 a^{2} c \,d^{2} e +6 a \,b^{2} d^{2} e -12 a b c \,d^{3}+b^{3} d^{3}\right )}{3 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(287\) |
gosper | \(-\frac {2 \left (12 a b c \,e^{3} x^{3}-24 a \,c^{2} d \,e^{2} x^{3}-e^{3} x^{3} b^{3}-6 b^{2} c d \,e^{2} x^{3}+24 b \,c^{2} d^{2} e \,x^{3}-16 d^{3} c^{3} x^{3}+24 a^{2} c \,e^{3} x^{2}+6 x^{2} a \,b^{2} e^{3}-36 a b c d \,e^{2} x^{2}-9 x^{2} b^{3} d \,e^{2}+36 b^{2} c \,d^{2} e \,x^{2}-24 b \,c^{2} d^{3} x^{2}+24 a^{2} b \,e^{3} x -36 x a \,b^{2} d \,e^{2}+36 a b c \,d^{2} e x -24 a \,c^{2} d^{3} x +9 b^{3} d^{2} e x -6 b^{2} c \,d^{3} x +16 e^{3} a^{3}-24 a^{2} b d \,e^{2}+24 a^{2} c \,d^{2} e +6 a \,b^{2} d^{2} e -12 a b c \,d^{3}+b^{3} d^{3}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\) | \(296\) |
orering | \(-\frac {2 \left (12 a b c \,e^{3} x^{3}-24 a \,c^{2} d \,e^{2} x^{3}-e^{3} x^{3} b^{3}-6 b^{2} c d \,e^{2} x^{3}+24 b \,c^{2} d^{2} e \,x^{3}-16 d^{3} c^{3} x^{3}+24 a^{2} c \,e^{3} x^{2}+6 x^{2} a \,b^{2} e^{3}-36 a b c d \,e^{2} x^{2}-9 x^{2} b^{3} d \,e^{2}+36 b^{2} c \,d^{2} e \,x^{2}-24 b \,c^{2} d^{3} x^{2}+24 a^{2} b \,e^{3} x -36 x a \,b^{2} d \,e^{2}+36 a b c \,d^{2} e x -24 a \,c^{2} d^{3} x +9 b^{3} d^{2} e x -6 b^{2} c \,d^{3} x +16 e^{3} a^{3}-24 a^{2} b d \,e^{2}+24 a^{2} c \,d^{2} e +6 a \,b^{2} d^{2} e -12 a b c \,d^{3}+b^{3} d^{3}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\) | \(296\) |
default | \(d^{3} \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )+e^{3} \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{2 c}+\frac {2 a \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{c}\right )+3 d \,e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )+3 d^{2} e \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )\) | \(675\) |
Input:
int((e*x+d)^3/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(12*a*b*c*e^3*x^3-24*a*c^2*d*e^2*x^3-b^3*e^3*x^3-6*b^2*c*d*e^2*x^3+24 *b*c^2*d^2*e*x^3-16*c^3*d^3*x^3+24*a^2*c*e^3*x^2+6*a*b^2*e^3*x^2-36*a*b*c* d*e^2*x^2-9*b^3*d*e^2*x^2+36*b^2*c*d^2*e*x^2-24*b*c^2*d^3*x^2+24*a^2*b*e^3 *x-36*a*b^2*d*e^2*x+36*a*b*c*d^2*e*x-24*a*c^2*d^3*x+9*b^3*d^2*e*x-6*b^2*c* d^3*x+16*a^3*e^3-24*a^2*b*d*e^2+24*a^2*c*d^2*e+6*a*b^2*d^2*e-12*a*b*c*d^3+ b^3*d^3)/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (110) = 220\).
Time = 0.57 (sec) , antiderivative size = 366, normalized size of antiderivative = 3.10 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (24 \, a^{2} b d e^{2} - 16 \, a^{3} e^{3} - {\left (b^{3} - 12 \, a b c\right )} d^{3} - 6 \, {\left (a b^{2} + 4 \, a^{2} c\right )} d^{2} e + {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d e^{2} + {\left (b^{3} - 12 \, a b c\right )} e^{3}\right )} x^{3} + 3 \, {\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, {\left (b^{3} + 4 \, a b c\right )} d e^{2} - 2 \, {\left (a b^{2} + 4 \, a^{2} c\right )} e^{3}\right )} x^{2} + 3 \, {\left (12 \, a b^{2} d e^{2} - 8 \, a^{2} b e^{3} + 2 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d^{3} - 3 \, {\left (b^{3} + 4 \, a b c\right )} d^{2} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \] Input:
integrate((e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
2/3*(24*a^2*b*d*e^2 - 16*a^3*e^3 - (b^3 - 12*a*b*c)*d^3 - 6*(a*b^2 + 4*a^2 *c)*d^2*e + (16*c^3*d^3 - 24*b*c^2*d^2*e + 6*(b^2*c + 4*a*c^2)*d*e^2 + (b^ 3 - 12*a*b*c)*e^3)*x^3 + 3*(8*b*c^2*d^3 - 12*b^2*c*d^2*e + 3*(b^3 + 4*a*b* c)*d*e^2 - 2*(a*b^2 + 4*a^2*c)*e^3)*x^2 + 3*(12*a*b^2*d*e^2 - 8*a^2*b*e^3 + 2*(b^2*c + 4*a*c^2)*d^3 - 3*(b^3 + 4*a*b*c)*d^2*e)*x)*sqrt(c*x^2 + b*x + a)/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2* c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)
Timed out. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**3/(c*x**2+b*x+a)**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^3/(c*x^2+b*x+a)^(5/2),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. 333 vs. \(2 (110) = 220\).
Time = 0.22 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.82 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (\frac {{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + 24 \, a c^{2} d e^{2} + b^{3} e^{3} - 12 \, a b c e^{3}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2} + 12 \, a b c d e^{2} - 2 \, a b^{2} e^{3} - 8 \, a^{2} c e^{3}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c d^{3} + 8 \, a c^{2} d^{3} - 3 \, b^{3} d^{2} e - 12 \, a b c d^{2} e + 12 \, a b^{2} d e^{2} - 8 \, a^{2} b e^{3}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {b^{3} d^{3} - 12 \, a b c d^{3} + 6 \, a b^{2} d^{2} e + 24 \, a^{2} c d^{2} e - 24 \, a^{2} b d e^{2} + 16 \, a^{3} e^{3}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \] Input:
integrate((e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
2/3*((((16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + 24*a*c^2*d*e^2 + b^3 *e^3 - 12*a*b*c*e^3)*x/(b^4 - 8*a*b^2*c + 16*a^2*c^2) + 3*(8*b*c^2*d^3 - 1 2*b^2*c*d^2*e + 3*b^3*d*e^2 + 12*a*b*c*d*e^2 - 2*a*b^2*e^3 - 8*a^2*c*e^3)/ (b^4 - 8*a*b^2*c + 16*a^2*c^2))*x + 3*(2*b^2*c*d^3 + 8*a*c^2*d^3 - 3*b^3*d ^2*e - 12*a*b*c*d^2*e + 12*a*b^2*d*e^2 - 8*a^2*b*e^3)/(b^4 - 8*a*b^2*c + 1 6*a^2*c^2))*x - (b^3*d^3 - 12*a*b*c*d^3 + 6*a*b^2*d^2*e + 24*a^2*c*d^2*e - 24*a^2*b*d*e^2 + 16*a^3*e^3)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))/(c*x^2 + b*x + a)^(3/2)
Time = 6.27 (sec) , antiderivative size = 528, normalized size of antiderivative = 4.47 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2\,a\,b^4\,e^3+2\,b^5\,e^3\,x-2\,b^4\,e^3\,\left (c\,x^2+b\,x+a\right )+16\,a^3\,c^2\,e^3-2\,b^3\,c^2\,d^3-12\,a^2\,b^2\,c\,e^3-48\,a^2\,c^3\,d^2\,e-4\,b^2\,c^3\,d^3\,x-48\,a^2\,c^2\,e^3\,\left (c\,x^2+b\,x+a\right )+8\,a\,b\,c^3\,d^3+16\,a\,c^4\,d^3\,x+16\,b\,c^3\,d^3\,\left (c\,x^2+b\,x+a\right )+32\,c^4\,d^3\,x\,\left (c\,x^2+b\,x+a\right )-6\,a\,b^3\,c\,d\,e^2-14\,a\,b^3\,c\,e^3\,x-6\,b^4\,c\,d\,e^2\,x+12\,a\,b^2\,c\,e^3\,\left (c\,x^2+b\,x+a\right )+6\,b^3\,c\,d\,e^2\,\left (c\,x^2+b\,x+a\right )+2\,b^3\,c\,e^3\,x\,\left (c\,x^2+b\,x+a\right )+12\,a\,b^2\,c^2\,d^2\,e+24\,a^2\,b\,c^2\,d\,e^2+24\,a^2\,b\,c^2\,e^3\,x-48\,a^2\,c^3\,d\,e^2\,x+6\,b^3\,c^2\,d^2\,e\,x-24\,b^2\,c^2\,d^2\,e\,\left (c\,x^2+b\,x+a\right )+12\,b^2\,c^2\,d\,e^2\,x\,\left (c\,x^2+b\,x+a\right )-24\,a\,b\,c^3\,d^2\,e\,x+24\,a\,b\,c^2\,d\,e^2\,\left (c\,x^2+b\,x+a\right )-24\,a\,b\,c^2\,e^3\,x\,\left (c\,x^2+b\,x+a\right )+48\,a\,c^3\,d\,e^2\,x\,\left (c\,x^2+b\,x+a\right )-48\,b\,c^3\,d^2\,e\,x\,\left (c\,x^2+b\,x+a\right )+36\,a\,b^2\,c^2\,d\,e^2\,x}{\left (48\,a^2\,c^4-24\,a\,b^2\,c^3+3\,b^4\,c^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \] Input:
int((d + e*x)^3/(a + b*x + c*x^2)^(5/2),x)
Output:
(2*a*b^4*e^3 + 2*b^5*e^3*x - 2*b^4*e^3*(a + b*x + c*x^2) + 16*a^3*c^2*e^3 - 2*b^3*c^2*d^3 - 12*a^2*b^2*c*e^3 - 48*a^2*c^3*d^2*e - 4*b^2*c^3*d^3*x - 48*a^2*c^2*e^3*(a + b*x + c*x^2) + 8*a*b*c^3*d^3 + 16*a*c^4*d^3*x + 16*b*c ^3*d^3*(a + b*x + c*x^2) + 32*c^4*d^3*x*(a + b*x + c*x^2) - 6*a*b^3*c*d*e^ 2 - 14*a*b^3*c*e^3*x - 6*b^4*c*d*e^2*x + 12*a*b^2*c*e^3*(a + b*x + c*x^2) + 6*b^3*c*d*e^2*(a + b*x + c*x^2) + 2*b^3*c*e^3*x*(a + b*x + c*x^2) + 12*a *b^2*c^2*d^2*e + 24*a^2*b*c^2*d*e^2 + 24*a^2*b*c^2*e^3*x - 48*a^2*c^3*d*e^ 2*x + 6*b^3*c^2*d^2*e*x - 24*b^2*c^2*d^2*e*(a + b*x + c*x^2) + 12*b^2*c^2* d*e^2*x*(a + b*x + c*x^2) - 24*a*b*c^3*d^2*e*x + 24*a*b*c^2*d*e^2*(a + b*x + c*x^2) - 24*a*b*c^2*e^3*x*(a + b*x + c*x^2) + 48*a*c^3*d*e^2*x*(a + b*x + c*x^2) - 48*b*c^3*d^2*e*x*(a + b*x + c*x^2) + 36*a*b^2*c^2*d*e^2*x)/((4 8*a^2*c^4 + 3*b^4*c^2 - 24*a*b^2*c^3)*(a + b*x + c*x^2)^(3/2))
Time = 0.97 (sec) , antiderivative size = 1237, normalized size of antiderivative = 10.48 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^3/(c*x^2+b*x+a)^(5/2),x)
Output:
(2*( - 16*sqrt(a + b*x + c*x**2)*a**3*c**2*e**3 + 24*sqrt(a + b*x + c*x**2 )*a**2*b*c**2*d*e**2 - 24*sqrt(a + b*x + c*x**2)*a**2*b*c**2*e**3*x - 24*s qrt(a + b*x + c*x**2)*a**2*c**3*d**2*e - 24*sqrt(a + b*x + c*x**2)*a**2*c* *3*e**3*x**2 - 6*sqrt(a + b*x + c*x**2)*a*b**2*c**2*d**2*e + 36*sqrt(a + b *x + c*x**2)*a*b**2*c**2*d*e**2*x - 6*sqrt(a + b*x + c*x**2)*a*b**2*c**2*e **3*x**2 + 12*sqrt(a + b*x + c*x**2)*a*b*c**3*d**3 - 36*sqrt(a + b*x + c*x **2)*a*b*c**3*d**2*e*x + 36*sqrt(a + b*x + c*x**2)*a*b*c**3*d*e**2*x**2 - 12*sqrt(a + b*x + c*x**2)*a*b*c**3*e**3*x**3 + 24*sqrt(a + b*x + c*x**2)*a *c**4*d**3*x + 24*sqrt(a + b*x + c*x**2)*a*c**4*d*e**2*x**3 - sqrt(a + b*x + c*x**2)*b**3*c**2*d**3 - 9*sqrt(a + b*x + c*x**2)*b**3*c**2*d**2*e*x + 9*sqrt(a + b*x + c*x**2)*b**3*c**2*d*e**2*x**2 + sqrt(a + b*x + c*x**2)*b* *3*c**2*e**3*x**3 + 6*sqrt(a + b*x + c*x**2)*b**2*c**3*d**3*x - 36*sqrt(a + b*x + c*x**2)*b**2*c**3*d**2*e*x**2 + 6*sqrt(a + b*x + c*x**2)*b**2*c**3 *d*e**2*x**3 + 24*sqrt(a + b*x + c*x**2)*b*c**4*d**3*x**2 - 24*sqrt(a + b* x + c*x**2)*b*c**4*d**2*e*x**3 + 16*sqrt(a + b*x + c*x**2)*c**5*d**3*x**3 - 12*sqrt(c)*a**3*b*c*e**3 + 24*sqrt(c)*a**3*c**2*d*e**2 + 5*sqrt(c)*a**2* b**3*e**3 - 18*sqrt(c)*a**2*b**2*c*d*e**2 - 24*sqrt(c)*a**2*b**2*c*e**3*x + 24*sqrt(c)*a**2*b*c**2*d**2*e + 48*sqrt(c)*a**2*b*c**2*d*e**2*x - 24*sqr t(c)*a**2*b*c**2*e**3*x**2 - 16*sqrt(c)*a**2*c**3*d**3 + 48*sqrt(c)*a**2*c **3*d*e**2*x**2 + 10*sqrt(c)*a*b**4*e**3*x - 36*sqrt(c)*a*b**3*c*d*e**2...