Integrand size = 37, antiderivative size = 169 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{d+e x} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{3 e}-\frac {\left (c d^2-a e^2\right )^2 \sqrt [4]{-\frac {c d e (a e+c d x) (d+e x)}{\left (c d^2-a e^2\right )^2}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {a e^2+c d (d+2 e x)}{c d^2-a e^2}\right )\right |2\right )}{\sqrt {2} c d e^2 \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:
2/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/e-1/2*(-a*e^2+c*d^2)^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))*2^(1/2)/c/d/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 = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{d+e x} \, dx=\frac {4 (a e+c d x)^2 \sqrt [4]{\frac {c d (d+e x)}{c d^2-a e^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{4},\frac {11}{4},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{7 c d \sqrt [4]{(a e+c d x) (d+e x)}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/4)/(d + e*x),x]
Output:
(4*(a*e + c*d*x)^2*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(1/4)*Hypergeometric2 F1[1/4, 7/4, 11/4, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(7*c*d*((a*e + c *d*x)*(d + e*x))^(1/4))
Time = 0.75 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.93, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {1138, 60, 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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1138 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \int \frac {(a e+c d x)^{3/4}}{\sqrt [4]{\frac {e x}{d}+1}}dx}{d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {2 d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}{3 e}-\frac {\left (c d^2-a e^2\right ) \int \frac {1}{\sqrt [4]{a e+c d x} \sqrt [4]{\frac {e x}{d}+1}}dx}{2 e}\right )}{d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {2 d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}{3 e}-\frac {2 \left (c d^2-a e^2\right ) \int \frac {\sqrt {a e+c d x}}{\sqrt [4]{-\frac {a e^2}{c d^2}+\frac {(a e+c d x) e}{c d^2}+1}}d\sqrt [4]{a e+c d x}}{c d e}\right )}{d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 839 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {2 d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}{3 e}-\frac {2 \left (c d^2-a e^2\right ) \left (\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}-\frac {1}{2} \left (1-\frac {a e^2}{c d^2}\right ) \int \frac {\sqrt {a e+c d x}}{\left (-\frac {a e^2}{c d^2}+\frac {(a e+c d x) e}{c d^2}+1\right )^{5/4}}d\sqrt [4]{a e+c d x}\right )}{c d e}\right )}{d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 813 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {2 d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}{3 e}-\frac {2 \left (c d^2-a e^2\right ) \left (\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}-\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} \int \frac {1}{(a e+c d x)^{3/4} \left (\frac {c d^2-a e^2}{e (a e+c d x)}+1\right )^{5/4}}d\sqrt [4]{a e+c d x}}{2 e \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{c d e}\right )}{d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {2 d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}{3 e}-\frac {2 \left (c d^2-a e^2\right ) \left (\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} \int \frac {1}{\sqrt [4]{a e+c d x} \left (\frac {\left (c d^2-a e^2\right ) (a e+c d x)}{e}+1\right )^{5/4}}d\frac {1}{\sqrt [4]{a e+c d x}}}{2 e \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}+\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{c d e}\right )}{d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {2 d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}{3 e}-\frac {2 \left (c d^2-a e^2\right ) \left (\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} \int \frac {1}{\left (\frac {\sqrt {a e+c d x} \left (c d^2-a e^2\right )}{e}+1\right )^{5/4}}d\sqrt {a e+c d x}}{4 e \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}+\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{c d e}\right )}{d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
| \(\Big \downarrow \) 212 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4} \left (\frac {2 d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}{3 e}-\frac {2 \left (c d^2-a e^2\right ) \left (\frac {c d^2 \left (1-\frac {a e^2}{c d^2}\right ) \sqrt [4]{a e+c d x} \sqrt [4]{\frac {c d^2-a e^2}{e (a e+c d x)}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {c d^2-a e^2} \sqrt {a e+c d x}}{\sqrt {e}}\right )\right |2\right )}{2 \sqrt {e} \sqrt {c d^2-a e^2} \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}+\frac {(a e+c d x)^{3/4}}{2 \sqrt [4]{-\frac {a e^2}{c d^2}+\frac {e (a e+c d x)}{c d^2}+1}}\right )}{c d e}\right )}{d \left (\frac {e x}{d}+1\right )^{3/4} (a e+c d x)^{3/4}}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/4)/(d + e*x),x]
Output:
((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/4)*((2*d*(a*e + c*d*x)^(3/4)*( 1 + (e*x)/d)^(3/4))/(3*e) - (2*(c*d^2 - a*e^2)*((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[ArcTan[(Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d*x])/Sqrt[e]]/2, 2] )/(2*Sqrt[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*e)))/(d*(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 + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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]))
\[\int \frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{4}}}{e x +d}d x\]
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/4)/(e*x+d),x)
Output:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/4)/(e*x+d),x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{d+e x} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{4}}}{e x + d} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/(e*x+d),x, algorithm="fr icas")
Output:
integral((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/4)/(e*x + d), x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{d+e x} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{4}}}{d + e x}\, dx \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/4)/(e*x+d),x)
Output:
Integral(((d + e*x)*(a*e + c*d*x))**(3/4)/(d + e*x), x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{d+e x} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{4}}}{e x + d} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/(e*x+d),x, algorithm="ma xima")
Output:
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/4)/(e*x + d), x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{d+e x} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{4}}}{e x + d} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/(e*x+d),x, algorithm="gi ac")
Output:
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/4)/(e*x + d), x)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{d+e x} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/4}}{d+e\,x} \,d x \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/4)/(d + e*x),x)
Output:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/4)/(d + e*x), x)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{d+e x} \, dx=\frac {4 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} a e -3 \left (\int \frac {\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} x}{a c d \,e^{3} x^{2}+c^{2} d^{3} e \,x^{2}+a^{2} e^{4} x +2 a c \,d^{2} e^{2} x +c^{2} d^{4} x +a^{2} d \,e^{3}+a c \,d^{3} e}d x \right ) a^{2} c d \,e^{4}+3 \left (\int \frac {\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} x}{a c d \,e^{3} x^{2}+c^{2} d^{3} e \,x^{2}+a^{2} e^{4} x +2 a c \,d^{2} e^{2} x +c^{2} d^{4} x +a^{2} d \,e^{3}+a c \,d^{3} e}d x \right ) c^{3} d^{5}}{3 a \,e^{2}+3 c \,d^{2}} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/(e*x+d),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*e**3 + a**2*e**4*x + a*c*d**3*e + 2*a*c*d**2*e**2*x + a*c*d*e**3*x**2 + c**2*d**4*x + c**2*d**3 *e*x**2),x)*a**2*c*d*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*e**3 + a**2*e**4*x + a*c*d**3*e + 2*a*c*d**2*e**2*x + a*c*d*e**3*x**2 + c**2*d**4*x + c**2*d**3*e*x**2),x)*c**3*d**5)/(3*(a*e* *2 + c*d**2))