Integrand size = 28, antiderivative size = 480 \[ \int \frac {(e+f x)^3 \sqrt {c+d x^3}}{a+b x^3} \, dx=\frac {2 e f^2 \sqrt {c+d x^3}}{b}+\frac {2 f^3 x \sqrt {c+d x^3}}{5 b}+\frac {(b c-a d) \left (b e^3-a f^3\right ) x \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a b^2 \sqrt {c+d x^3}}+\frac {3 e^2 f x^2 \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {2}{3},1,-\frac {1}{2},\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a \sqrt {1+\frac {d x^3}{c}}}-\frac {2 \sqrt {b c-a d} e f^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{b^{3/2}}+\frac {2 \sqrt {2+\sqrt {3}} \left (5 b d e^3+3 b c f^3-5 a d f^3\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} b^2 \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
2*e*f^2*(d*x^3+c)^(1/2)/b+2/5*f^3*x*(d*x^3+c)^(1/2)/b+(-a*d+b*c)*(-a*f^3+b *e^3)*x*(1+d*x^3/c)^(1/2)*AppellF1(1/3,1,1/2,4/3,-b*x^3/a,-d*x^3/c)/a/b^2/ (d*x^3+c)^(1/2)+3/2*e^2*f*x^2*(d*x^3+c)^(1/2)*AppellF1(2/3,1,-1/2,5/3,-b*x ^3/a,-d*x^3/c)/a/(1+d*x^3/c)^(1/2)-2*(-a*d+b*c)^(1/2)*e*f^2*arctanh(b^(1/2 )*(d*x^3+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(3/2)+2/15*(1/2*6^(1/2)+1/2*2^(1/2)) *(-5*a*d*f^3+3*b*c*f^3+5*b*d*e^3)*(c^(1/3)+d^(1/3)*x)*((c^(2/3)-c^(1/3)*d^ (1/3)*x+d^(2/3)*x^2)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)*EllipticF((( 1-3^(1/2))*c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x),I*3^(1/2)+2* I)*3^(3/4)/b^2/d^(1/3)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d ^(1/3)*x)^2)^(1/2)/(d*x^3+c)^(1/2)
Time = 7.90 (sec) , antiderivative size = 715, normalized size of antiderivative = 1.49 \[ \int \frac {(e+f x)^3 \sqrt {c+d x^3}}{a+b x^3} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^3*Sqrt[c + d*x^3])/(a + b*x^3),x]
Output:
((40*c*e*f^2)/b + (8*c*f^3*x)/b + (40*d*e*f^2*x^3)/b + (8*d*f^3*x^4)/b + ( 30*c*e^2*f*x^2*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c) , -((b*x^3)/a)])/a + (20*(b*c - a*d)*e*f^2*x^3*Sqrt[1 + (d*x^3)/c]*AppellF 1[1, 1/2, 1, 2, -((d*x^3)/c), -((b*x^3)/a)])/(a*b) + (5*d*e^3*x^4*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/a + (3 *c*f^3*x^4*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -( (b*x^3)/a)])/a - (5*d*f^3*x^4*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/ 3, -((d*x^3)/c), -((b*x^3)/a)])/b - (160*a*c^2*e^3*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)])/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3 )/c), -((b*x^3)/a)]))) + (64*a^2*c^2*f^3*x*AppellF1[1/3, 1/2, 1, 4/3, -((d *x^3)/c), -((b*x^3)/a)])/(b*(a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -(( d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -( (b*x^3)/a)]))) + (12*d*e^2*f*x^5*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/a)/(20*Sqrt[c + d*x^3])
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.
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x^3} \left (-a f^3+b e^3+3 b e^2 f x+3 b e f^2 x^2\right )}{b \left (a+b x^3\right )}+\frac {f^3 \sqrt {c+d x^3}}{b}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt {c+d x^3} (e+f x)^3}{a+b x^3}dx\) |
Input:
Int[((e + f*x)^3*Sqrt[c + d*x^3])/(a + b*x^3),x]
Output:
$Aborted
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 3.45 (sec) , antiderivative size = 1304, normalized size of antiderivative = 2.72
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1304\) |
| default | \(\text {Expression too large to display}\) | \(1565\) |
| risch | \(\text {Expression too large to display}\) | \(1836\) |
Input:
int((f*x+e)^3*(d*x^3+c)^(1/2)/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
2/5*f^3*x*(d*x^3+c)^(1/2)/b+2*e*f^2*(d*x^3+c)^(1/2)/b-2/3*I*(-(a*d*f^3-b*c *f^3-b*d*e^3)/b^2-2/5*c*f^3/b)*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^ 2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*( (x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/ 3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^( 1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1 /2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/ 3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2) /d*(-c*d^2)^(1/3)))^(1/2))-2*I*e^2*f/b*3^(1/2)*(-c*d^2)^(1/3)*(I*(x+1/2/d* (-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^( 1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^ 2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3 ))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3) +1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2 )^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I *3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^ (1/3)))^(1/2))+1/d*(-c*d^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^ 2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),( I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2) ^(1/3)))^(1/2)))+1/3*I/b^2/d^2*2^(1/2)*sum((3*_alpha^2*a*b*d*e*f^2-3*_a...
Timed out. \[ \int \frac {(e+f x)^3 \sqrt {c+d x^3}}{a+b x^3} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*(d*x^3+c)^(1/2)/(b*x^3+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(e+f x)^3 \sqrt {c+d x^3}}{a+b x^3} \, dx=\int \frac {\sqrt {c + d x^{3}} \left (e + f x\right )^{3}}{a + b x^{3}}\, dx \] Input:
integrate((f*x+e)**3*(d*x**3+c)**(1/2)/(b*x**3+a),x)
Output:
Integral(sqrt(c + d*x**3)*(e + f*x)**3/(a + b*x**3), x)
\[ \int \frac {(e+f x)^3 \sqrt {c+d x^3}}{a+b x^3} \, dx=\int { \frac {\sqrt {d x^{3} + c} {\left (f x + e\right )}^{3}}{b x^{3} + a} \,d x } \] Input:
integrate((f*x+e)^3*(d*x^3+c)^(1/2)/(b*x^3+a),x, algorithm="maxima")
Output:
integrate(sqrt(d*x^3 + c)*(f*x + e)^3/(b*x^3 + a), x)
\[ \int \frac {(e+f x)^3 \sqrt {c+d x^3}}{a+b x^3} \, dx=\int { \frac {\sqrt {d x^{3} + c} {\left (f x + e\right )}^{3}}{b x^{3} + a} \,d x } \] Input:
integrate((f*x+e)^3*(d*x^3+c)^(1/2)/(b*x^3+a),x, algorithm="giac")
Output:
integrate(sqrt(d*x^3 + c)*(f*x + e)^3/(b*x^3 + a), x)
Timed out. \[ \int \frac {(e+f x)^3 \sqrt {c+d x^3}}{a+b x^3} \, dx=\int \frac {{\left (e+f\,x\right )}^3\,\sqrt {d\,x^3+c}}{b\,x^3+a} \,d x \] Input:
int(((e + f*x)^3*(c + d*x^3)^(1/2))/(a + b*x^3),x)
Output:
int(((e + f*x)^3*(c + d*x^3)^(1/2))/(a + b*x^3), x)
\[ \int \frac {(e+f x)^3 \sqrt {c+d x^3}}{a+b x^3} \, dx=\frac {2 \sqrt {d \,x^{3}+c}\, a d \,f^{3} x +10 \sqrt {d \,x^{3}+c}\, b c e \,f^{2}-2 \left (\int \frac {\sqrt {d \,x^{3}+c}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} c d \,f^{3}+5 \left (\int \frac {\sqrt {d \,x^{3}+c}}{b \,x^{3}+a}d x \right ) a b d \,e^{3}+15 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a b \,d^{2} e \,f^{2}-15 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b^{2} c d e \,f^{2}-5 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} d^{2} f^{3}+3 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a b c d \,f^{3}+15 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{b \,x^{3}+a}d x \right ) a b d \,e^{2} f}{5 a b d} \] Input:
int((f*x+e)^3*(d*x^3+c)^(1/2)/(b*x^3+a),x)
Output:
(2*sqrt(c + d*x**3)*a*d*f**3*x + 10*sqrt(c + d*x**3)*b*c*e*f**2 - 2*int(sq rt(c + d*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a**2*c*d*f**3 + 5 *int(sqrt(c + d*x**3)/(a + b*x**3),x)*a*b*d*e**3 + 15*int((sqrt(c + d*x**3 )*x**5)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a*b*d**2*e*f**2 - 15*int ((sqrt(c + d*x**3)*x**5)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*b**2*c* d*e*f**2 - 5*int((sqrt(c + d*x**3)*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b*d* x**6),x)*a**2*d**2*f**3 + 3*int((sqrt(c + d*x**3)*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a*b*c*d*f**3 + 15*int((sqrt(c + d*x**3)*x)/(a + b* x**3),x)*a*b*d*e**2*f)/(5*a*b*d)