Integrand size = 30, antiderivative size = 422 \[ \int \frac {\left (a+b x^3\right )^{5/2}}{\left (c+d x^3\right ) \left (e+f x^3\right )} \, dx=\frac {2 b^2 x \sqrt {a+b x^3}}{5 d f}-\frac {(b c-a d)^3 x \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c d^2 (d e-c f) \sqrt {a+b x^3}}+\frac {(b e-a f)^3 x \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{e f^2 (d e-c f) \sqrt {a+b x^3}}+\frac {2 \sqrt {2+\sqrt {3}} b^{5/3} (13 a d f-5 b (d e+c f)) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} d^2 f^2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
2/5*b^2*x*(b*x^3+a)^(1/2)/d/f-(-a*d+b*c)^3*x*(1+b*x^3/a)^(1/2)*AppellF1(1/ 3,1/2,1,4/3,-b*x^3/a,-d*x^3/c)/c/d^2/(-c*f+d*e)/(b*x^3+a)^(1/2)+(-a*f+b*e) ^3*x*(1+b*x^3/a)^(1/2)*AppellF1(1/3,1/2,1,4/3,-b*x^3/a,-f*x^3/e)/e/f^2/(-c *f+d*e)/(b*x^3+a)^(1/2)+2/15*(1/2*6^(1/2)+1/2*2^(1/2))*b^(5/3)*(13*a*d*f-5 *b*(c*f+d*e))*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2) /((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b ^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)*3^(3/4)/d^2/f^2/( a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)/(b*x^ 3+a)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(12526\) vs. \(2(422)=844\).
Time = 16.63 (sec) , antiderivative size = 12526, normalized size of antiderivative = 29.68 \[ \int \frac {\left (a+b x^3\right )^{5/2}}{\left (c+d x^3\right ) \left (e+f x^3\right )} \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*x^3)^(5/2)/((c + d*x^3)*(e + f*x^3)),x]
Output:
Result too large to show
Time = 1.35 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.89, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1028, 937, 936, 1028, 937, 936, 1028, 937, 936, 1029, 937, 936}
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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{5/2}}{\left (c+d x^3\right ) \left (e+f x^3\right )} \, dx\) |
| \(\Big \downarrow \) 1028 |
| \(\displaystyle \frac {b \int \frac {\left (b x^3+a\right )^{3/2}}{f x^3+e}dx}{d}-\frac {(b c-a d) \int \frac {\left (b x^3+a\right )^{3/2}}{\left (d x^3+c\right ) \left (f x^3+e\right )}dx}{d}\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle \frac {a b \sqrt {a+b x^3} \int \frac {\left (\frac {b x^3}{a}+1\right )^{3/2}}{f x^3+e}dx}{d \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \int \frac {\left (b x^3+a\right )^{3/2}}{\left (d x^3+c\right ) \left (f x^3+e\right )}dx}{d}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle \frac {a b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \int \frac {\left (b x^3+a\right )^{3/2}}{\left (d x^3+c\right ) \left (f x^3+e\right )}dx}{d}\) |
| \(\Big \downarrow \) 1028 |
| \(\displaystyle \frac {a b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b \int \frac {\sqrt {b x^3+a}}{f x^3+e}dx}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^3+a}}{\left (d x^3+c\right ) \left (f x^3+e\right )}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle \frac {a b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b \sqrt {a+b x^3} \int \frac {\sqrt {\frac {b x^3}{a}+1}}{f x^3+e}dx}{d \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \int \frac {\sqrt {b x^3+a}}{\left (d x^3+c\right ) \left (f x^3+e\right )}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle \frac {a b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \int \frac {\sqrt {b x^3+a}}{\left (d x^3+c\right ) \left (f x^3+e\right )}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 1028 |
| \(\displaystyle \frac {a b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b \int \frac {1}{\sqrt {b x^3+a} \left (f x^3+e\right )}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^3+a} \left (d x^3+c\right ) \left (f x^3+e\right )}dx}{d}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle \frac {a b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b \sqrt {\frac {b x^3}{a}+1} \int \frac {1}{\sqrt {\frac {b x^3}{a}+1} \left (f x^3+e\right )}dx}{d \sqrt {a+b x^3}}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^3+a} \left (d x^3+c\right ) \left (f x^3+e\right )}dx}{d}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle \frac {a b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {a+b x^3}}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^3+a} \left (d x^3+c\right ) \left (f x^3+e\right )}dx}{d}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 1029 |
| \(\displaystyle \frac {a b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {a+b x^3}}-\frac {(b c-a d) \left (\frac {d \int \frac {1}{\sqrt {b x^3+a} \left (d x^3+c\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^3+a} \left (f x^3+e\right )}dx}{d e-c f}\right )}{d}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle \frac {a b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {a+b x^3}}-\frac {(b c-a d) \left (\frac {d \sqrt {\frac {b x^3}{a}+1} \int \frac {1}{\sqrt {\frac {b x^3}{a}+1} \left (d x^3+c\right )}dx}{\sqrt {a+b x^3} (d e-c f)}-\frac {f \sqrt {\frac {b x^3}{a}+1} \int \frac {1}{\sqrt {\frac {b x^3}{a}+1} \left (f x^3+e\right )}dx}{\sqrt {a+b x^3} (d e-c f)}\right )}{d}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle \frac {a b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b x \sqrt {a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {\frac {b x^3}{a}+1}}-\frac {(b c-a d) \left (\frac {b x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{d e \sqrt {a+b x^3}}-\frac {(b c-a d) \left (\frac {d x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \sqrt {a+b x^3} (d e-c f)}-\frac {f x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {f x^3}{e}\right )}{e \sqrt {a+b x^3} (d e-c f)}\right )}{d}\right )}{d}\right )}{d}\) |
Input:
Int[(a + b*x^3)^(5/2)/((c + d*x^3)*(e + f*x^3)),x]
Output:
(a*b*x*Sqrt[a + b*x^3]*AppellF1[1/3, -3/2, 1, 4/3, -((b*x^3)/a), -((f*x^3) /e)])/(d*e*Sqrt[1 + (b*x^3)/a]) - ((b*c - a*d)*((b*x*Sqrt[a + b*x^3]*Appel lF1[1/3, -1/2, 1, 4/3, -((b*x^3)/a), -((f*x^3)/e)])/(d*e*Sqrt[1 + (b*x^3)/ a]) - ((b*c - a*d)*((b*x*Sqrt[1 + (b*x^3)/a]*AppellF1[1/3, 1/2, 1, 4/3, -( (b*x^3)/a), -((f*x^3)/e)])/(d*e*Sqrt[a + b*x^3]) - ((b*c - a*d)*((d*x*Sqrt [1 + (b*x^3)/a]*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(c *(d*e - c*f)*Sqrt[a + b*x^3]) - (f*x*Sqrt[1 + (b*x^3)/a]*AppellF1[1/3, 1/2 , 1, 4/3, -((b*x^3)/a), -((f*x^3)/e)])/(e*(d*e - c*f)*Sqrt[a + b*x^3])))/d ))/d))/d
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) ], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q }, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_ .)*(x_)^(n_))^(r_), x_Symbol] :> Simp[d/b Int[(a + b*x^n)^(p + 1)*(c + d* x^n)^(q - 1)*(e + f*x^n)^r, x], x] + Simp[(b*c - a*d)/b Int[(a + b*x^n)^p *(c + d*x^n)^(q - 1)*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, n, r }, x] && ILtQ[p, 0] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_ .)*(x_)^(n_))^(r_), x_Symbol] :> Simp[b/(b*c - a*d) Int[(a + b*x^n)^p*(c + d*x^n)^(q + 1)*(e + f*x^n)^r, x], x] - Simp[d/(b*c - a*d) Int[(a + b*x^ n)^(p + 1)*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && ILtQ[p, 0] && LeQ[q, -1]
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 3.60 (sec) , antiderivative size = 1290, normalized size of antiderivative = 3.06
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1290\) |
| default | \(\text {Expression too large to display}\) | \(1769\) |
| elliptic | \(\text {Expression too large to display}\) | \(4213\) |
Input:
int((b*x^3+a)^(5/2)/(d*x^3+c)/(f*x^3+e),x,method=_RETURNVERBOSE)
Output:
2/5*b^2*x*(b*x^3+a)^(1/2)/d/f+1/5/d/f*(-2/3*I*b*(13*a*d*f-5*b*c*f-5*b*d*e) /d/f*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a *b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2 /b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a* b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2) /(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^ (1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b ^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+5 /3*I*d/f*(a^3*f^3-3*a^2*b*e*f^2+3*a*b^2*e^2*f-b^3*e^3)/(c*f-d*e)/b^2*2^(1/ 2)*sum(1/_alpha^2/(a*f-b*e)*(-a*b^2)^(1/3)*(1/2*I*b*(2*x+1/b*(-I*3^(1/2)*( -a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(-a*b^2)^(1 /3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1/2*I*b*(2*x+1/ b*(I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(b*x^3+ a)^(1/2)*(I*(-a*b^2)^(1/3)*_alpha*3^(1/2)*b-I*3^(1/2)*(-a*b^2)^(2/3)+2*_al pha^2*b^2-(-a*b^2)^(1/3)*_alpha*b-(-a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*( I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^ 2)^(1/3))^(1/2),-1/2/b*f*(2*I*3^(1/2)*(-a*b^2)^(1/3)*_alpha^2*b-I*3^(1/2)* (-a*b^2)^(2/3)*_alpha+I*3^(1/2)*a*b-3*(-a*b^2)^(2/3)*_alpha-3*a*b)/(a*f-b* e),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a* b^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*f+e))-5/3*I/d*f*(a^3*d^3-3*a^2*b...
Timed out. \[ \int \frac {\left (a+b x^3\right )^{5/2}}{\left (c+d x^3\right ) \left (e+f x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x^3+a)^(5/2)/(d*x^3+c)/(f*x^3+e),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^3\right )^{5/2}}{\left (c+d x^3\right ) \left (e+f x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x**3+a)**(5/2)/(d*x**3+c)/(f*x**3+e),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^3\right )^{5/2}}{\left (c+d x^3\right ) \left (e+f x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {5}{2}}}{{\left (d x^{3} + c\right )} {\left (f x^{3} + e\right )}} \,d x } \] Input:
integrate((b*x^3+a)^(5/2)/(d*x^3+c)/(f*x^3+e),x, algorithm="maxima")
Output:
integrate((b*x^3 + a)^(5/2)/((d*x^3 + c)*(f*x^3 + e)), x)
\[ \int \frac {\left (a+b x^3\right )^{5/2}}{\left (c+d x^3\right ) \left (e+f x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {5}{2}}}{{\left (d x^{3} + c\right )} {\left (f x^{3} + e\right )}} \,d x } \] Input:
integrate((b*x^3+a)^(5/2)/(d*x^3+c)/(f*x^3+e),x, algorithm="giac")
Output:
integrate((b*x^3 + a)^(5/2)/((d*x^3 + c)*(f*x^3 + e)), x)
Timed out. \[ \int \frac {\left (a+b x^3\right )^{5/2}}{\left (c+d x^3\right ) \left (e+f x^3\right )} \, dx=\text {Hanged} \] Input:
int((a + b*x^3)^(5/2)/((c + d*x^3)*(e + f*x^3)),x)
Output:
\text{Hanged}
\[ \int \frac {\left (a+b x^3\right )^{5/2}}{\left (c+d x^3\right ) \left (e+f x^3\right )} \, dx=\frac {2 \sqrt {b \,x^{3}+a}\, b^{2} x +5 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b d f \,x^{9}+a d f \,x^{6}+b c f \,x^{6}+b d e \,x^{6}+a c f \,x^{3}+a d e \,x^{3}+b c e \,x^{3}+a c e}d x \right ) a^{3} d f -2 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b d f \,x^{9}+a d f \,x^{6}+b c f \,x^{6}+b d e \,x^{6}+a c f \,x^{3}+a d e \,x^{3}+b c e \,x^{3}+a c e}d x \right ) a \,b^{2} c e +13 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{6}}{b d f \,x^{9}+a d f \,x^{6}+b c f \,x^{6}+b d e \,x^{6}+a c f \,x^{3}+a d e \,x^{3}+b c e \,x^{3}+a c e}d x \right ) a \,b^{2} d f -5 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{6}}{b d f \,x^{9}+a d f \,x^{6}+b c f \,x^{6}+b d e \,x^{6}+a c f \,x^{3}+a d e \,x^{3}+b c e \,x^{3}+a c e}d x \right ) b^{3} c f -5 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{6}}{b d f \,x^{9}+a d f \,x^{6}+b c f \,x^{6}+b d e \,x^{6}+a c f \,x^{3}+a d e \,x^{3}+b c e \,x^{3}+a c e}d x \right ) b^{3} d e +15 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{3}}{b d f \,x^{9}+a d f \,x^{6}+b c f \,x^{6}+b d e \,x^{6}+a c f \,x^{3}+a d e \,x^{3}+b c e \,x^{3}+a c e}d x \right ) a^{2} b d f -2 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{3}}{b d f \,x^{9}+a d f \,x^{6}+b c f \,x^{6}+b d e \,x^{6}+a c f \,x^{3}+a d e \,x^{3}+b c e \,x^{3}+a c e}d x \right ) a \,b^{2} c f -2 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{3}}{b d f \,x^{9}+a d f \,x^{6}+b c f \,x^{6}+b d e \,x^{6}+a c f \,x^{3}+a d e \,x^{3}+b c e \,x^{3}+a c e}d x \right ) a \,b^{2} d e -5 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x^{3}}{b d f \,x^{9}+a d f \,x^{6}+b c f \,x^{6}+b d e \,x^{6}+a c f \,x^{3}+a d e \,x^{3}+b c e \,x^{3}+a c e}d x \right ) b^{3} c e}{5 d f} \] Input:
int((b*x^3+a)^(5/2)/(d*x^3+c)/(f*x^3+e),x)
Output:
(2*sqrt(a + b*x**3)*b**2*x + 5*int(sqrt(a + b*x**3)/(a*c*e + a*c*f*x**3 + a*d*e*x**3 + a*d*f*x**6 + b*c*e*x**3 + b*c*f*x**6 + b*d*e*x**6 + b*d*f*x** 9),x)*a**3*d*f - 2*int(sqrt(a + b*x**3)/(a*c*e + a*c*f*x**3 + a*d*e*x**3 + a*d*f*x**6 + b*c*e*x**3 + b*c*f*x**6 + b*d*e*x**6 + b*d*f*x**9),x)*a*b**2 *c*e + 13*int((sqrt(a + b*x**3)*x**6)/(a*c*e + a*c*f*x**3 + a*d*e*x**3 + a *d*f*x**6 + b*c*e*x**3 + b*c*f*x**6 + b*d*e*x**6 + b*d*f*x**9),x)*a*b**2*d *f - 5*int((sqrt(a + b*x**3)*x**6)/(a*c*e + a*c*f*x**3 + a*d*e*x**3 + a*d* f*x**6 + b*c*e*x**3 + b*c*f*x**6 + b*d*e*x**6 + b*d*f*x**9),x)*b**3*c*f - 5*int((sqrt(a + b*x**3)*x**6)/(a*c*e + a*c*f*x**3 + a*d*e*x**3 + a*d*f*x** 6 + b*c*e*x**3 + b*c*f*x**6 + b*d*e*x**6 + b*d*f*x**9),x)*b**3*d*e + 15*in t((sqrt(a + b*x**3)*x**3)/(a*c*e + a*c*f*x**3 + a*d*e*x**3 + a*d*f*x**6 + b*c*e*x**3 + b*c*f*x**6 + b*d*e*x**6 + b*d*f*x**9),x)*a**2*b*d*f - 2*int(( sqrt(a + b*x**3)*x**3)/(a*c*e + a*c*f*x**3 + a*d*e*x**3 + a*d*f*x**6 + b*c *e*x**3 + b*c*f*x**6 + b*d*e*x**6 + b*d*f*x**9),x)*a*b**2*c*f - 2*int((sqr t(a + b*x**3)*x**3)/(a*c*e + a*c*f*x**3 + a*d*e*x**3 + a*d*f*x**6 + b*c*e* x**3 + b*c*f*x**6 + b*d*e*x**6 + b*d*f*x**9),x)*a*b**2*d*e - 5*int((sqrt(a + b*x**3)*x**3)/(a*c*e + a*c*f*x**3 + a*d*e*x**3 + a*d*f*x**6 + b*c*e*x** 3 + b*c*f*x**6 + b*d*e*x**6 + b*d*f*x**9),x)*b**3*c*e)/(5*d*f)