Integrand size = 37, antiderivative size = 303 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=-\frac {4}{\left (c d^2-a e^2\right ) (d+e x)^2 \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {40 e \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{9 \left (c d^2-a e^2\right )^2 (d+e x)^3}-\frac {16 c d e \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{3 \left (c d^2-a e^2\right )^3 (d+e x)^2}+\frac {32 c^2 d^2 \sqrt {e} \sqrt {a e+c d x} \sqrt [4]{\frac {c d (d+e x)}{e (a e+c d x)}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {c d^2-a e^2}}{\sqrt {e} \sqrt {a e+c d x}}\right )\right |2\right )}{3 \left (c d^2-a e^2\right )^{7/2} \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:
-4/(-a*e^2+c*d^2)/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/4)-40/9*e *(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/(-a*e^2+c*d^2)^2/(e*x+d)^3-16/3*c *d*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/(-a*e^2+c*d^2)^3/(e*x+d)^2+32 /3*c^2*d^2*e^(1/2)*(c*d*x+a*e)^(1/2)*(c*d*(e*x+d)/e/(c*d*x+a*e))^(1/4)*Ell ipticE(sin(1/2*arctan((-a*e^2+c*d^2)^(1/2)/e^(1/2)/(c*d*x+a*e)^(1/2))),2^( 1/2))/(-a*e^2+c*d^2)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.33 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=-\frac {4 c^2 d^2 \sqrt [4]{\frac {c d (d+e x)}{c d^2-a e^2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {13}{4},\frac {3}{4},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{\left (c d^2-a e^2\right )^3 \sqrt [4]{(a e+c d x) (d+e x)}} \] Input:
Integrate[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/4)),x]
Output:
(-4*c^2*d^2*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(1/4)*Hypergeometric2F1[-1/4 , 13/4, 3/4, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)^3*((a *e + c*d*x)*(d + e*x))^(1/4))
Time = 0.64 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.71, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {1138, 61, 61, 61, 61, 73, 839, 813, 858, 807, 212}
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.
| \(\displaystyle \int \frac {1}{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/4}} \, dx\) |
| \(\Big \downarrow \) 1138 |
| \(\displaystyle \frac {\sqrt [4]{\frac {e x}{d}+1} \sqrt [4]{a e+c d x} \int \frac {1}{(a e+c d x)^{5/4} \left (\frac {e x}{d}+1\right )^{13/4}}dx}{d^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {\sqrt [4]{\frac {e x}{d}+1} \sqrt [4]{a e+c d x} \left (-\frac {10 e \int \frac {1}{\sqrt [4]{a e+c d x} \left (\frac {e x}{d}+1\right )^{13/4}}dx}{c d^2-a e^2}-\frac {4}{\left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right ) \sqrt [4]{a e+c d x}}\right )}{d^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {\sqrt [4]{\frac {e x}{d}+1} \sqrt [4]{a e+c d x} \left (-\frac {10 e \left (\frac {2 c d \int \frac {1}{\sqrt [4]{a e+c d x} \left (\frac {e x}{d}+1\right )^{9/4}}dx}{3 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{9 \left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{c d^2-a e^2}-\frac {4}{\left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right ) \sqrt [4]{a e+c d x}}\right )}{d^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {\sqrt [4]{\frac {e x}{d}+1} \sqrt [4]{a e+c d x} \left (-\frac {10 e \left (\frac {2 c d \left (\frac {2 c d \int \frac {1}{\sqrt [4]{a e+c d x} \left (\frac {e x}{d}+1\right )^{5/4}}dx}{5 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{5 \left (\frac {e x}{d}+1\right )^{5/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{3 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{9 \left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{c d^2-a e^2}-\frac {4}{\left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right ) \sqrt [4]{a e+c d x}}\right )}{d^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {\sqrt [4]{\frac {e x}{d}+1} \sqrt [4]{a e+c d x} \left (-\frac {10 e \left (\frac {2 c d \left (\frac {2 c d \left (\frac {4 d (a e+c d x)^{3/4}}{\sqrt [4]{\frac {e x}{d}+1} \left (c d^2-a e^2\right )}-\frac {2 c d^2 \int \frac {1}{\sqrt [4]{a e+c d x} \sqrt [4]{\frac {e x}{d}+1}}dx}{c d^2-a e^2}\right )}{5 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{5 \left (\frac {e x}{d}+1\right )^{5/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{3 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{9 \left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{c d^2-a e^2}-\frac {4}{\left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right ) \sqrt [4]{a e+c d x}}\right )}{d^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sqrt [4]{\frac {e x}{d}+1} \sqrt [4]{a e+c d x} \left (-\frac {10 e \left (\frac {2 c d \left (\frac {2 c d \left (\frac {4 d (a e+c d x)^{3/4}}{\sqrt [4]{\frac {e x}{d}+1} \left (c d^2-a e^2\right )}-\frac {8 d \int \frac {\sqrt {a e+c d x}}{\sqrt [4]{-\frac {a e^2}{c d^2}+\frac {(a e+c d x) e}{c d^2}+1}}d\sqrt [4]{a e+c d x}}{c d^2-a e^2}\right )}{5 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{5 \left (\frac {e x}{d}+1\right )^{5/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{3 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{9 \left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{c d^2-a e^2}-\frac {4}{\left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right ) \sqrt [4]{a e+c d x}}\right )}{d^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 839 |
| \(\displaystyle \frac {\sqrt [4]{\frac {e x}{d}+1} \sqrt [4]{a e+c d x} \left (-\frac {10 e \left (\frac {2 c d \left (\frac {2 c d \left (\frac {4 d (a e+c d x)^{3/4}}{\sqrt [4]{\frac {e x}{d}+1} \left (c d^2-a e^2\right )}-\frac {8 d \left (\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}-\frac {1}{2} \left (1-\frac {a e^2}{c d^2}\right ) \int \frac {\sqrt {a e+c d x}}{\left (-\frac {a e^2}{c d^2}+\frac {(a e+c d x) e}{c d^2}+1\right )^{5/4}}d\sqrt [4]{a e+c d x}\right )}{c d^2-a e^2}\right )}{5 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{5 \left (\frac {e x}{d}+1\right )^{5/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{3 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{9 \left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{c d^2-a e^2}-\frac {4}{\left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right ) \sqrt [4]{a e+c d x}}\right )}{d^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 813 |
| \(\displaystyle \frac {\sqrt [4]{\frac {e x}{d}+1} \sqrt [4]{a e+c d x} \left (-\frac {10 e \left (\frac {2 c d \left (\frac {2 c d \left (\frac {4 d (a e+c d x)^{3/4}}{\sqrt [4]{\frac {e x}{d}+1} \left (c d^2-a e^2\right )}-\frac {8 d \left (\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}-\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} \int \frac {1}{(a e+c d x)^{3/4} \left (\frac {c d^2-a e^2}{e (a e+c d x)}+1\right )^{5/4}}d\sqrt [4]{a e+c d x}}{2 e \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{c d^2-a e^2}\right )}{5 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{5 \left (\frac {e x}{d}+1\right )^{5/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{3 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{9 \left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{c d^2-a e^2}-\frac {4}{\left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right ) \sqrt [4]{a e+c d x}}\right )}{d^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle \frac {\sqrt [4]{\frac {e x}{d}+1} \sqrt [4]{a e+c d x} \left (-\frac {10 e \left (\frac {2 c d \left (\frac {2 c d \left (\frac {4 d (a e+c d x)^{3/4}}{\sqrt [4]{\frac {e x}{d}+1} \left (c d^2-a e^2\right )}-\frac {8 d \left (\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} \int \frac {1}{\sqrt [4]{a e+c d x} \left (\frac {\left (c d^2-a e^2\right ) (a e+c d x)}{e}+1\right )^{5/4}}d\frac {1}{\sqrt [4]{a e+c d x}}}{2 e \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}+\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{c d^2-a e^2}\right )}{5 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{5 \left (\frac {e x}{d}+1\right )^{5/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{3 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{9 \left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{c d^2-a e^2}-\frac {4}{\left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right ) \sqrt [4]{a e+c d x}}\right )}{d^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {\sqrt [4]{\frac {e x}{d}+1} \sqrt [4]{a e+c d x} \left (-\frac {10 e \left (\frac {2 c d \left (\frac {2 c d \left (\frac {4 d (a e+c d x)^{3/4}}{\sqrt [4]{\frac {e x}{d}+1} \left (c d^2-a e^2\right )}-\frac {8 d \left (\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} \int \frac {1}{\left (\frac {\sqrt {a e+c d x} \left (c d^2-a e^2\right )}{e}+1\right )^{5/4}}d\sqrt {a e+c d x}}{4 e \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}+\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{c d^2-a e^2}\right )}{5 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{5 \left (\frac {e x}{d}+1\right )^{5/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{3 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{9 \left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{c d^2-a e^2}-\frac {4}{\left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right ) \sqrt [4]{a e+c d x}}\right )}{d^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 212 |
| \(\displaystyle \frac {\sqrt [4]{\frac {e x}{d}+1} \sqrt [4]{a e+c d x} \left (-\frac {10 e \left (\frac {2 c d \left (\frac {2 c d \left (\frac {4 d (a e+c d x)^{3/4}}{\sqrt [4]{\frac {e x}{d}+1} \left (c d^2-a e^2\right )}-\frac {8 d \left (\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {c d^2-a e^2} \sqrt {a e+c d x}}{\sqrt {e}}\right )\right |2\right )}{2 \sqrt {e} \sqrt {c d^2-a e^2} \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}+\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{c d^2-a e^2}\right )}{5 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{5 \left (\frac {e x}{d}+1\right )^{5/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{3 \left (c d-\frac {a e^2}{d}\right )}+\frac {4 (a e+c d x)^{3/4}}{9 \left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right )}\right )}{c d^2-a e^2}-\frac {4}{\left (\frac {e x}{d}+1\right )^{9/4} \left (c d-\frac {a e^2}{d}\right ) \sqrt [4]{a e+c d x}}\right )}{d^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
Input:
Int[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/4)),x]
Output:
((a*e + c*d*x)^(1/4)*(1 + (e*x)/d)^(1/4)*(-4/((c*d - (a*e^2)/d)*(a*e + c*d *x)^(1/4)*(1 + (e*x)/d)^(9/4)) - (10*e*((4*(a*e + c*d*x)^(3/4))/(9*(c*d - (a*e^2)/d)*(1 + (e*x)/d)^(9/4)) + (2*c*d*((4*(a*e + c*d*x)^(3/4))/(5*(c*d - (a*e^2)/d)*(1 + (e*x)/d)^(5/4)) + (2*c*d*((4*d*(a*e + c*d*x)^(3/4))/((c* d^2 - a*e^2)*(1 + (e*x)/d)^(1/4)) - (8*d*((a*e + c*d*x)^(3/4)/(2*(1 - (a*e ^2)/(c*d^2) + (e*(a*e + c*d*x))/(c*d^2))^(1/4)) + (c*d^2*(1 - (a*e^2)/(c*d ^2))*(a*e + c*d*x)^(1/4)*(1 + (c*d^2 - a*e^2)/(e*(a*e + c*d*x)))^(1/4)*Ell ipticE[ArcTan[(Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d*x])/Sqrt[e]]/2, 2])/(2*S qrt[e]*Sqrt[c*d^2 - a*e^2]*(1 - (a*e^2)/(c*d^2) + (e*(a*e + c*d*x))/(c*d^2 ))^(1/4))))/(c*d^2 - a*e^2)))/(5*(c*d - (a*e^2)/d))))/(3*(c*d - (a*e^2)/d) )))/(c*d^2 - a*e^2)))/(d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/4))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Simp[x*((1 + a/(b*x^4) )^(1/4)/(b*(a + b*x^4)^(1/4))) Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(1/4), x_Symbol] :> Simp[x^3/(2*(a + b*x^4 )^(1/4)), x] - Simp[a/2 Int[x^2/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b} , x] && PosQ[b/a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[d^m*((a + b*x + c*x^2)^FracPart[p]/((1 + e*(x/d))^FracPart[p] *(a/d + (c*x)/e)^FracPart[p])) Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^ p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[m] || GtQ[d, 0]) && !(IGtQ[m, 0] && (IntegerQ[3*p] || Integer Q[4*p]))
\[\int \frac {1}{\left (e x +d \right )^{2} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{4}}}d x\]
Input:
int(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/4),x)
Output:
int(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/4),x)
\[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x, algorithm ="fricas")
Output:
integral((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/4)/(c^2*d^2*e^4*x^6 + a^2*d^4*e^2 + 2*(2*c^2*d^3*e^3 + a*c*d*e^5)*x^5 + (6*c^2*d^4*e^2 + 8*a*c*d ^2*e^4 + a^2*e^6)*x^4 + 4*(c^2*d^5*e + 3*a*c*d^3*e^3 + a^2*d*e^5)*x^3 + (c ^2*d^6 + 8*a*c*d^4*e^2 + 6*a^2*d^2*e^4)*x^2 + 2*(a*c*d^5*e + 2*a^2*d^3*e^3 )*x), x)
\[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{4}} \left (d + e x\right )^{2}}\, dx \] Input:
integrate(1/(e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/4),x)
Output:
Integral(1/(((d + e*x)*(a*e + c*d*x))**(5/4)*(d + e*x)**2), x)
\[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x, algorithm ="maxima")
Output:
integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/4)*(e*x + d)^2), x )
\[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x, algorithm ="giac")
Output:
integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/4)*(e*x + d)^2), x )
Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/4}} \,d x \] Input:
int(1/((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/4)),x)
Output:
int(1/((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/4)), x)
\[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int \frac {1}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} a \,d^{3} e +3 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} a \,d^{2} e^{2} x +3 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} a d \,e^{3} x^{2}+\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} a \,e^{4} x^{3}+\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} c \,d^{4} x +3 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} c \,d^{3} e \,x^{2}+3 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} c \,d^{2} e^{2} x^{3}+\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} c d \,e^{3} x^{4}}d x \] Input:
int(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x)
Output:
int(1/((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*a*d**3*e + 3*(a*d *e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*a*d**2*e**2*x + 3*(a*d*e + a *e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*a*d*e**3*x**2 + (a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*a*e**4*x**3 + (a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*c*d**4*x + 3*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x* *2)**(1/4)*c*d**3*e*x**2 + 3*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**( 1/4)*c*d**2*e**2*x**3 + (a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)* c*d*e**3*x**4),x)