\(\int \frac {a+b x^3+c x^6}{(d+e x^3)^{7/2}} \, dx\) [15]

Optimal result

Integrand size = 24, antiderivative size = 349 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{7/2}} \, dx=\frac {2 \left (c d^2-b d e+a e^2\right ) x}{15 d e^2 \left (d+e x^3\right )^{5/2}}-\frac {2 \left (17 c d^2-2 b d e-13 a e^2\right ) x}{135 d^2 e^2 \left (d+e x^3\right )^{3/2}}+\frac {2 \left (16 c d^2+14 b d e+91 a e^2\right ) x}{405 d^3 e^2 \sqrt {d+e x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (16 c d^2+14 b d e+91 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt {3}\right )}{405 \sqrt [4]{3} d^3 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \] Output:

2/15*(a*e^2-b*d*e+c*d^2)*x/d/e^2/(e*x^3+d)^(5/2)-2/135*(-13*a*e^2-2*b*d*e+ 
17*c*d^2)*x/d^2/e^2/(e*x^3+d)^(3/2)+2/405*(91*a*e^2+14*b*d*e+16*c*d^2)*x/d 
^3/e^2/(e*x^3+d)^(1/2)+2/1215*(1/2*6^(1/2)+1/2*2^(1/2))*(91*a*e^2+14*b*d*e 
+16*c*d^2)*(d^(1/3)+e^(1/3)*x)*((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(( 
1+3^(1/2))*d^(1/3)+e^(1/3)*x)^2)^(1/2)*EllipticF(((1-3^(1/2))*d^(1/3)+e^(1 
/3)*x)/((1+3^(1/2))*d^(1/3)+e^(1/3)*x),I*3^(1/2)+2*I)*3^(3/4)/d^3/e^(7/3)/ 
(d^(1/3)*(d^(1/3)+e^(1/3)*x)/((1+3^(1/2))*d^(1/3)+e^(1/3)*x)^2)^(1/2)/(e*x 
^3+d)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.48 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{7/2}} \, dx=\frac {2 x \left (c d^2 \left (-8 d^2-19 d e x^3+16 e^2 x^6\right )+e \left (b d \left (-7 d^2+34 d e x^3+14 e^2 x^6\right )+a e \left (157 d^2+221 d e x^3+91 e^2 x^6\right )\right )\right )+\left (16 c d^2+7 e (2 b d+13 a e)\right ) x \left (d+e x^3\right )^2 \sqrt {1+\frac {e x^3}{d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {e x^3}{d}\right )}{405 d^3 e^2 \left (d+e x^3\right )^{5/2}} \] Input:

Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^(7/2),x]
 

Output:

(2*x*(c*d^2*(-8*d^2 - 19*d*e*x^3 + 16*e^2*x^6) + e*(b*d*(-7*d^2 + 34*d*e*x 
^3 + 14*e^2*x^6) + a*e*(157*d^2 + 221*d*e*x^3 + 91*e^2*x^6))) + (16*c*d^2 
+ 7*e*(2*b*d + 13*a*e))*x*(d + e*x^3)^2*Sqrt[1 + (e*x^3)/d]*Hypergeometric 
2F1[1/3, 1/2, 4/3, -((e*x^3)/d)])/(405*d^3*e^2*(d + e*x^3)^(5/2))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1739, 27, 910, 749, 759}

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.

\(\displaystyle \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1739

\(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{15 d e^2 \left (d+e x^3\right )^{5/2}}-\frac {2 \int \frac {-15 c d e x^3+2 c d^2-e (2 b d+13 a e)}{2 \left (e x^3+d\right )^{5/2}}dx}{15 d e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{15 d e^2 \left (d+e x^3\right )^{5/2}}-\frac {\int \frac {-15 c d e x^3+2 c d^2-13 a e^2-2 b d e}{\left (e x^3+d\right )^{5/2}}dx}{15 d e^2}\)

\(\Big \downarrow \) 910

\(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{15 d e^2 \left (d+e x^3\right )^{5/2}}-\frac {\frac {2 x \left (-13 a e^2-2 b d e+17 c d^2\right )}{9 d \left (d+e x^3\right )^{3/2}}-\frac {\left (91 a e^2+14 b d e+16 c d^2\right ) \int \frac {1}{\left (e x^3+d\right )^{3/2}}dx}{9 d}}{15 d e^2}\)

\(\Big \downarrow \) 749

\(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{15 d e^2 \left (d+e x^3\right )^{5/2}}-\frac {\frac {2 x \left (-13 a e^2-2 b d e+17 c d^2\right )}{9 d \left (d+e x^3\right )^{3/2}}-\frac {\left (91 a e^2+14 b d e+16 c d^2\right ) \left (\frac {\int \frac {1}{\sqrt {e x^3+d}}dx}{3 d}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{9 d}}{15 d e^2}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{15 d e^2 \left (d+e x^3\right )^{5/2}}-\frac {\frac {2 x \left (-13 a e^2-2 b d e+17 c d^2\right )}{9 d \left (d+e x^3\right )^{3/2}}-\frac {\left (91 a e^2+14 b d e+16 c d^2\right ) \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{9 d}}{15 d e^2}\)

Input:

Int[(a + b*x^3 + c*x^6)/(d + e*x^3)^(7/2),x]
 

Output:

(2*(c*d^2 - b*d*e + a*e^2)*x)/(15*d*e^2*(d + e*x^3)^(5/2)) - ((2*(17*c*d^2 
 - 2*b*d*e - 13*a*e^2)*x)/(9*d*(d + e*x^3)^(3/2)) - ((16*c*d^2 + 14*b*d*e 
+ 91*a*e^2)*((2*x)/(3*d*Sqrt[d + e*x^3]) + (2*Sqrt[2 + Sqrt[3]]*(d^(1/3) + 
 e^(1/3)*x)*Sqrt[(d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2)/((1 + Sqrt[3] 
)*d^(1/3) + e^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*d^(1/3) + e^(1/3 
)*x)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*d*e 
^(1/3)*Sqrt[(d^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + Sqrt[3])*d^(1/3) + e^(1/ 
3)*x)^2]*Sqrt[d + e*x^3])))/(9*d))/(15*d*e^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 749

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 
 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1))   Int[(a + b*x^ 
n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte 
gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
 

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 910

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - 
 b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1))   Int[(a + b*x^n)^(p + 1), x], x] /; 
FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ 
n + p, 0])
 

rule 1739

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ 
)), x_Symbol] :> Simp[(-(c*d^2 - b*d*e + a*e^2))*x*((d + e*x^n)^(q + 1)/(d* 
e^2*n*(q + 1))), x] + Simp[1/(n*(q + 1)*d*e^2)   Int[(d + e*x^n)^(q + 1)*Si 
mp[c*d^2 - b*d*e + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x], x] 
/; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && N 
eQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[q, -1]
 

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.25

method	result	size
elliptic	\(\frac {2 x \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}{15 d \,e^{5} \left (x^{3}+\frac {d}{e}\right )^{3}}+\frac {2 x \left (13 a \,e^{2}+2 b d e -17 c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}{135 d^{2} e^{4} \left (x^{3}+\frac {d}{e}\right )^{2}}+\frac {2 x \left (91 a \,e^{2}+14 b d e +16 c \,d^{2}\right )}{405 e^{2} d^{3} \sqrt {\left (x^{3}+\frac {d}{e}\right ) e}}-\frac {2 i \left (91 a \,e^{2}+14 b d e +16 c \,d^{2}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{1215 d^{3} e^{3} \sqrt {e \,x^{3}+d}}\)	\(437\)
default	\(\text {Expression too large to display}\)	\(1095\)

Input:

int((c*x^6+b*x^3+a)/(e*x^3+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

2/15*x/d/e^5*(a*e^2-b*d*e+c*d^2)*(e*x^3+d)^(1/2)/(x^3+d/e)^3+2/135*x/d^2*( 
13*a*e^2+2*b*d*e-17*c*d^2)/e^4*(e*x^3+d)^(1/2)/(x^3+d/e)^2+2/405/e^2*x/d^3 
*(91*a*e^2+14*b*d*e+16*c*d^2)/((x^3+d/e)*e)^(1/2)-2/1215*I*(91*a*e^2+14*b* 
d*e+16*c*d^2)/d^3/e^3*3^(1/2)*(-d*e^2)^(1/3)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/ 
2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2)*((x-1/e*(-d* 
e^2)^(1/3))/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(1/2)* 
(-I*(x+1/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d* 
e^2)^(1/3))^(1/2)/(e*x^3+d)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/e*(-d*e^ 
2)^(1/3)-1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2),( 
I*3^(1/2)/e*(-d*e^2)^(1/3)/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2) 
^(1/3)))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.77 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{7/2}} \, dx=\frac {2 \, {\left ({\left ({\left (16 \, c d^{2} e^{3} + 14 \, b d e^{4} + 91 \, a e^{5}\right )} x^{9} + 3 \, {\left (16 \, c d^{3} e^{2} + 14 \, b d^{2} e^{3} + 91 \, a d e^{4}\right )} x^{6} + 16 \, c d^{5} + 14 \, b d^{4} e + 91 \, a d^{3} e^{2} + 3 \, {\left (16 \, c d^{4} e + 14 \, b d^{3} e^{2} + 91 \, a d^{2} e^{3}\right )} x^{3}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (0, -\frac {4 \, d}{e}, x\right ) + {\left ({\left (16 \, c d^{2} e^{3} + 14 \, b d e^{4} + 91 \, a e^{5}\right )} x^{7} - {\left (19 \, c d^{3} e^{2} - 34 \, b d^{2} e^{3} - 221 \, a d e^{4}\right )} x^{4} - {\left (8 \, c d^{4} e + 7 \, b d^{3} e^{2} - 157 \, a d^{2} e^{3}\right )} x\right )} \sqrt {e x^{3} + d}\right )}}{405 \, {\left (d^{3} e^{6} x^{9} + 3 \, d^{4} e^{5} x^{6} + 3 \, d^{5} e^{4} x^{3} + d^{6} e^{3}\right )}} \] Input:

integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(7/2),x, algorithm="fricas")
 

Output:

2/405*(((16*c*d^2*e^3 + 14*b*d*e^4 + 91*a*e^5)*x^9 + 3*(16*c*d^3*e^2 + 14* 
b*d^2*e^3 + 91*a*d*e^4)*x^6 + 16*c*d^5 + 14*b*d^4*e + 91*a*d^3*e^2 + 3*(16 
*c*d^4*e + 14*b*d^3*e^2 + 91*a*d^2*e^3)*x^3)*sqrt(e)*weierstrassPInverse(0 
, -4*d/e, x) + ((16*c*d^2*e^3 + 14*b*d*e^4 + 91*a*e^5)*x^7 - (19*c*d^3*e^2 
 - 34*b*d^2*e^3 - 221*a*d*e^4)*x^4 - (8*c*d^4*e + 7*b*d^3*e^2 - 157*a*d^2* 
e^3)*x)*sqrt(e*x^3 + d))/(d^3*e^6*x^9 + 3*d^4*e^5*x^6 + 3*d^5*e^4*x^3 + d^ 
6*e^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((c*x**6+b*x**3+a)/(e*x**3+d)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{7/2}} \, dx=\int { \frac {c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(7/2),x, algorithm="maxima")
 

Output:

integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{7/2}} \, dx=\int { \frac {c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(7/2),x, algorithm="giac")
 

Output:

integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{7/2}} \, dx=\int \frac {c\,x^6+b\,x^3+a}{{\left (e\,x^3+d\right )}^{7/2}} \,d x \] Input:

int((a + b*x^3 + c*x^6)/(d + e*x^3)^(7/2),x)
 

Output:

int((a + b*x^3 + c*x^6)/(d + e*x^3)^(7/2), x)
 

Reduce [F]

\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{7/2}} \, dx=\frac {-14 \sqrt {e \,x^{3}+d}\, b e x -16 \sqrt {e \,x^{3}+d}\, c d x -26 \sqrt {e \,x^{3}+d}\, c e \,x^{4}+91 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) a \,d^{3} e^{2}+273 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) a \,d^{2} e^{3} x^{3}+273 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) a d \,e^{4} x^{6}+91 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) a \,e^{5} x^{9}+14 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) b \,d^{4} e +42 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) b \,d^{3} e^{2} x^{3}+42 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) b \,d^{2} e^{3} x^{6}+14 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) b d \,e^{4} x^{9}+16 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) c \,d^{5}+48 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) c \,d^{4} e \,x^{3}+48 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) c \,d^{3} e^{2} x^{6}+16 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{4} x^{12}+4 d \,e^{3} x^{9}+6 d^{2} e^{2} x^{6}+4 d^{3} e \,x^{3}+d^{4}}d x \right ) c \,d^{2} e^{3} x^{9}}{91 e^{2} \left (e^{3} x^{9}+3 d \,e^{2} x^{6}+3 d^{2} e \,x^{3}+d^{3}\right )} \] Input:

int((c*x^6+b*x^3+a)/(e*x^3+d)^(7/2),x)
 

Output:

( - 14*sqrt(d + e*x**3)*b*e*x - 16*sqrt(d + e*x**3)*c*d*x - 26*sqrt(d + e* 
x**3)*c*e*x**4 + 91*int(sqrt(d + e*x**3)/(d**4 + 4*d**3*e*x**3 + 6*d**2*e* 
*2*x**6 + 4*d*e**3*x**9 + e**4*x**12),x)*a*d**3*e**2 + 273*int(sqrt(d + e* 
x**3)/(d**4 + 4*d**3*e*x**3 + 6*d**2*e**2*x**6 + 4*d*e**3*x**9 + e**4*x**1 
2),x)*a*d**2*e**3*x**3 + 273*int(sqrt(d + e*x**3)/(d**4 + 4*d**3*e*x**3 + 
6*d**2*e**2*x**6 + 4*d*e**3*x**9 + e**4*x**12),x)*a*d*e**4*x**6 + 91*int(s 
qrt(d + e*x**3)/(d**4 + 4*d**3*e*x**3 + 6*d**2*e**2*x**6 + 4*d*e**3*x**9 + 
 e**4*x**12),x)*a*e**5*x**9 + 14*int(sqrt(d + e*x**3)/(d**4 + 4*d**3*e*x** 
3 + 6*d**2*e**2*x**6 + 4*d*e**3*x**9 + e**4*x**12),x)*b*d**4*e + 42*int(sq 
rt(d + e*x**3)/(d**4 + 4*d**3*e*x**3 + 6*d**2*e**2*x**6 + 4*d*e**3*x**9 + 
e**4*x**12),x)*b*d**3*e**2*x**3 + 42*int(sqrt(d + e*x**3)/(d**4 + 4*d**3*e 
*x**3 + 6*d**2*e**2*x**6 + 4*d*e**3*x**9 + e**4*x**12),x)*b*d**2*e**3*x**6 
 + 14*int(sqrt(d + e*x**3)/(d**4 + 4*d**3*e*x**3 + 6*d**2*e**2*x**6 + 4*d* 
e**3*x**9 + e**4*x**12),x)*b*d*e**4*x**9 + 16*int(sqrt(d + e*x**3)/(d**4 + 
 4*d**3*e*x**3 + 6*d**2*e**2*x**6 + 4*d*e**3*x**9 + e**4*x**12),x)*c*d**5 
+ 48*int(sqrt(d + e*x**3)/(d**4 + 4*d**3*e*x**3 + 6*d**2*e**2*x**6 + 4*d*e 
**3*x**9 + e**4*x**12),x)*c*d**4*e*x**3 + 48*int(sqrt(d + e*x**3)/(d**4 + 
4*d**3*e*x**3 + 6*d**2*e**2*x**6 + 4*d*e**3*x**9 + e**4*x**12),x)*c*d**3*e 
**2*x**6 + 16*int(sqrt(d + e*x**3)/(d**4 + 4*d**3*e*x**3 + 6*d**2*e**2*x** 
6 + 4*d*e**3*x**9 + e**4*x**12),x)*c*d**2*e**3*x**9)/(91*e**2*(d**3 + 3...