Integrand size = 37, antiderivative size = 166 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{(d+e x)^2} \, dx=-\frac {4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{e (d+e x)}+\frac {3 \sqrt {2} \left (c d^2-a e^2\right ) \sqrt [4]{-\frac {c d e (a e+c d x) (d+e x)}{\left (c d^2-a e^2\right )^2}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {a e^2+c d (d+2 e x)}{c d^2-a e^2}\right )\right |2\right )}{e^2 \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:
-4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/e/(e*x+d)+3*2^(1/2)*(-a*e^2+c*d ^2)*(-c*d*e*(c*d*x+a*e)*(e*x+d)/(-a*e^2+c*d^2)^2)^(1/4)*EllipticE(sin(1/2* arcsin((a*e^2+c*d*(2*e*x+d))/(-a*e^2+c*d^2))),2^(1/2))/e^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 = 106, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{(d+e x)^2} \, dx=\frac {4 c d (a e+c d x) ((a e+c d x) (d+e x))^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {7}{4},\frac {11}{4},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{7 \left (c d^2-a e^2\right )^2 \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{3/4}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/4)/(d + e*x)^2,x]
Output:
(4*c*d*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(3/4)*Hypergeometric2F1[5/4 , 7/4, 11/4, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(7*(c*d^2 - a*e^2)^2*( (c*d*(d + e*x))/(c*d^2 - a*e^2))^(3/4))
Time = 0.74 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.85, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {1138, 57, 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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1138 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \int \frac {(a e+c d x)^{3/4}}{\left (\frac {e x}{d}+1\right )^{5/4}}dx}{d^2 \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {3 c d^2 \int \frac {1}{\sqrt [4]{a e+c d x} \sqrt [4]{\frac {e x}{d}+1}}dx}{e}-\frac {4 d (a e+c d x)^{3/4}}{e \sqrt [4]{\frac {e x}{d}+1}}\right )}{d^2 \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {12 d \int \frac {\sqrt {a e+c d x}}{\sqrt [4]{-\frac {a e^2}{c d^2}+\frac {(a e+c d x) e}{c d^2}+1}}d\sqrt [4]{a e+c d x}}{e}-\frac {4 d (a e+c d x)^{3/4}}{e \sqrt [4]{\frac {e x}{d}+1}}\right )}{d^2 \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 839 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {12 d \left (\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}-\frac {1}{2} \left (1-\frac {a e^2}{c d^2}\right ) \int \frac {\sqrt {a e+c d x}}{\left (-\frac {a e^2}{c d^2}+\frac {(a e+c d x) e}{c d^2}+1\right )^{5/4}}d\sqrt [4]{a e+c d x}\right )}{e}-\frac {4 d (a e+c d x)^{3/4}}{e \sqrt [4]{\frac {e x}{d}+1}}\right )}{d^2 \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 813 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {12 d \left (\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}-\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} \int \frac {1}{(a e+c d x)^{3/4} \left (\frac {c d^2-a e^2}{e (a e+c d x)}+1\right )^{5/4}}d\sqrt [4]{a e+c d x}}{2 e \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{e}-\frac {4 d (a e+c d x)^{3/4}}{e \sqrt [4]{\frac {e x}{d}+1}}\right )}{d^2 \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {12 d \left (\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} \int \frac {1}{\sqrt [4]{a e+c d x} \left (\frac {\left (c d^2-a e^2\right ) (a e+c d x)}{e}+1\right )^{5/4}}d\frac {1}{\sqrt [4]{a e+c d x}}}{2 e \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}+\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{e}-\frac {4 d (a e+c d x)^{3/4}}{e \sqrt [4]{\frac {e x}{d}+1}}\right )}{d^2 \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {12 d \left (\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} \int \frac {1}{\left (\frac {\sqrt {a e+c d x} \left (c d^2-a e^2\right )}{e}+1\right )^{5/4}}d\sqrt {a e+c d x}}{4 e \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}+\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{e}-\frac {4 d (a e+c d x)^{3/4}}{e \sqrt [4]{\frac {e x}{d}+1}}\right )}{d^2 \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 212 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {12 d \left (\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {c d^2-a e^2} \sqrt {a e+c d x}}{\sqrt {e}}\right )\right |2\right )}{2 \sqrt {e} \sqrt {c d^2-a e^2} \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}+\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{e}-\frac {4 d (a e+c d x)^{3/4}}{e \sqrt [4]{\frac {e x}{d}+1}}\right )}{d^2 \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/4)/(d + e*x)^2,x]
Output:
((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/4)*((-4*d*(a*e + c*d*x)^(3/4)) /(e*(1 + (e*x)/d)^(1/4)) + (12*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)*EllipticE[A rcTan[(Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d*x])/Sqrt[e]]/2, 2])/(2*Sqrt[e]*S qrt[c*d^2 - a*e^2]*(1 - (a*e^2)/(c*d^2) + (e*(a*e + c*d*x))/(c*d^2))^(1/4) )))/e))/(d^2*(a*e + c*d*x)^(3/4)*(1 + (e*x)/d)^(3/4))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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]))
\[\int \frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{4}}}{\left (e x +d \right )^{2}}d x\]
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/4)/(e*x+d)^2,x)
Output:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/4)/(e*x+d)^2,x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{(d+e x)^2} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{4}}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/(e*x+d)^2,x, algorithm=" fricas")
Output:
integral((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/4)/(e^2*x^2 + 2*d*e*x + d^2), x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{(d+e x)^2} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{4}}}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/4)/(e*x+d)**2,x)
Output:
Integral(((d + e*x)*(a*e + c*d*x))**(3/4)/(d + e*x)**2, x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{(d+e x)^2} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{4}}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/(e*x+d)^2,x, algorithm=" maxima")
Output:
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/4)/(e*x + d)^2, x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{(d+e x)^2} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{4}}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/(e*x+d)^2,x, algorithm=" giac")
Output:
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/4)/(e*x + d)^2, x)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/4}}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/4)/(d + e*x)^2,x)
Output:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/4)/(d + e*x)^2, x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/(e*x+d)^2,x)
Output:
( - 4*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(3/4)*a*e + 3*int(((a*d* e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(3/4)*x)/(a**2*d**2*e**3 + 2*a**2*d *e**4*x + a**2*e**5*x**2 - 3*a*c*d**4*e - 5*a*c*d**3*e**2*x - a*c*d**2*e** 3*x**2 + a*c*d*e**4*x**3 - 3*c**2*d**5*x - 6*c**2*d**4*e*x**2 - 3*c**2*d** 3*e**2*x**3),x)*a**2*c*d**2*e**4 + 3*int(((a*d*e + a*e**2*x + c*d**2*x + c *d*e*x**2)**(3/4)*x)/(a**2*d**2*e**3 + 2*a**2*d*e**4*x + a**2*e**5*x**2 - 3*a*c*d**4*e - 5*a*c*d**3*e**2*x - a*c*d**2*e**3*x**2 + a*c*d*e**4*x**3 - 3*c**2*d**5*x - 6*c**2*d**4*e*x**2 - 3*c**2*d**3*e**2*x**3),x)*a**2*c*d*e* *5*x - 12*int(((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(3/4)*x)/(a**2* d**2*e**3 + 2*a**2*d*e**4*x + a**2*e**5*x**2 - 3*a*c*d**4*e - 5*a*c*d**3*e **2*x - a*c*d**2*e**3*x**2 + a*c*d*e**4*x**3 - 3*c**2*d**5*x - 6*c**2*d**4 *e*x**2 - 3*c**2*d**3*e**2*x**3),x)*a*c**2*d**4*e**2 - 12*int(((a*d*e + a* e**2*x + c*d**2*x + c*d*e*x**2)**(3/4)*x)/(a**2*d**2*e**3 + 2*a**2*d*e**4* x + a**2*e**5*x**2 - 3*a*c*d**4*e - 5*a*c*d**3*e**2*x - a*c*d**2*e**3*x**2 + a*c*d*e**4*x**3 - 3*c**2*d**5*x - 6*c**2*d**4*e*x**2 - 3*c**2*d**3*e**2 *x**3),x)*a*c**2*d**3*e**3*x + 9*int(((a*d*e + a*e**2*x + c*d**2*x + c*d*e *x**2)**(3/4)*x)/(a**2*d**2*e**3 + 2*a**2*d*e**4*x + a**2*e**5*x**2 - 3*a* c*d**4*e - 5*a*c*d**3*e**2*x - a*c*d**2*e**3*x**2 + a*c*d*e**4*x**3 - 3*c* *2*d**5*x - 6*c**2*d**4*e*x**2 - 3*c**2*d**3*e**2*x**3),x)*c**3*d**6 + 9*i nt(((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(3/4)*x)/(a**2*d**2*e**...