\(\int (e x)^{3/2} (a+b x^3)^p (c+d x^3) \, dx\) [406]

Optimal result

Integrand size = 24, antiderivative size = 107 \[ \int (e x)^{3/2} \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\frac {2 d (e x)^{5/2} \left (a+b x^3\right )^{1+p}}{b e (11+6 p)}+\frac {2 \left (c-\frac {5 a d}{11 b+6 b p}\right ) (e x)^{5/2} \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},-p,\frac {11}{6},-\frac {b x^3}{a}\right )}{5 e} \] Output:

2*d*(e*x)^(5/2)*(b*x^3+a)^(p+1)/b/e/(11+6*p)+2/5*(c-5*a*d/(6*b*p+11*b))*(e 
*x)^(5/2)*(b*x^3+a)^p*hypergeom([5/6, -p],[11/6],-b*x^3/a)/e/((1+b*x^3/a)^ 
p)
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.78 \[ \int (e x)^{3/2} \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\frac {2}{55} x (e x)^{3/2} \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \left (11 c \operatorname {Hypergeometric2F1}\left (\frac {5}{6},-p,\frac {11}{6},-\frac {b x^3}{a}\right )+5 d x^3 \operatorname {Hypergeometric2F1}\left (\frac {11}{6},-p,\frac {17}{6},-\frac {b x^3}{a}\right )\right ) \] Input:

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

Output:

(2*x*(e*x)^(3/2)*(a + b*x^3)^p*(11*c*Hypergeometric2F1[5/6, -p, 11/6, -((b 
*x^3)/a)] + 5*d*x^3*Hypergeometric2F1[11/6, -p, 17/6, -((b*x^3)/a)]))/(55* 
(1 + (b*x^3)/a)^p)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {959, 889, 888}

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.

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

\(\Big \downarrow \) 959

\(\displaystyle \left (c-\frac {5 a d}{6 b p+11 b}\right ) \int (e x)^{3/2} \left (b x^3+a\right )^pdx+\frac {2 d (e x)^{5/2} \left (a+b x^3\right )^{p+1}}{b e (6 p+11)}\)

\(\Big \downarrow \) 889

\(\displaystyle \left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} \left (c-\frac {5 a d}{6 b p+11 b}\right ) \int (e x)^{3/2} \left (\frac {b x^3}{a}+1\right )^pdx+\frac {2 d (e x)^{5/2} \left (a+b x^3\right )^{p+1}}{b e (6 p+11)}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {2 (e x)^{5/2} \left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} \left (c-\frac {5 a d}{6 b p+11 b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {5}{6},-p,\frac {11}{6},-\frac {b x^3}{a}\right )}{5 e}+\frac {2 d (e x)^{5/2} \left (a+b x^3\right )^{p+1}}{b e (6 p+11)}\)

Input:

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

Output:

(2*d*(e*x)^(5/2)*(a + b*x^3)^(1 + p))/(b*e*(11 + 6*p)) + (2*(c - (5*a*d)/( 
11*b + 6*b*p))*(e*x)^(5/2)*(a + b*x^3)^p*Hypergeometric2F1[5/6, -p, 11/6, 
-((b*x^3)/a)])/(5*e*(1 + (b*x^3)/a)^p)
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

rule 959

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p 
+ 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p 
 + 1) + 1))   Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, 
 n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

Maple [F]

\[\int \left (e x \right )^{\frac {3}{2}} \left (b \,x^{3}+a \right )^{p} \left (d \,x^{3}+c \right )d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d*e*x^4 + c*e*x)*sqrt(e*x)*(b*x^3 + a)^p, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^{3/2} \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\frac {2 \sqrt {e}\, e \left (6 \sqrt {x}\, \left (b \,x^{3}+a \right )^{p} a d p \,x^{2}+6 \sqrt {x}\, \left (b \,x^{3}+a \right )^{p} b c p \,x^{2}+11 \sqrt {x}\, \left (b \,x^{3}+a \right )^{p} b c \,x^{2}+6 \sqrt {x}\, \left (b \,x^{3}+a \right )^{p} b d p \,x^{5}+5 \sqrt {x}\, \left (b \,x^{3}+a \right )^{p} b d \,x^{5}-540 \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p} x}{36 b \,p^{2} x^{3}+96 b p \,x^{3}+55 b \,x^{3}+36 a \,p^{2}+96 a p +55 a}d x \right ) a^{2} d \,p^{3}-1440 \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p} x}{36 b \,p^{2} x^{3}+96 b p \,x^{3}+55 b \,x^{3}+36 a \,p^{2}+96 a p +55 a}d x \right ) a^{2} d \,p^{2}-825 \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p} x}{36 b \,p^{2} x^{3}+96 b p \,x^{3}+55 b \,x^{3}+36 a \,p^{2}+96 a p +55 a}d x \right ) a^{2} d p +648 \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p} x}{36 b \,p^{2} x^{3}+96 b p \,x^{3}+55 b \,x^{3}+36 a \,p^{2}+96 a p +55 a}d x \right ) a b c \,p^{4}+2916 \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p} x}{36 b \,p^{2} x^{3}+96 b p \,x^{3}+55 b \,x^{3}+36 a \,p^{2}+96 a p +55 a}d x \right ) a b c \,p^{3}+4158 \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p} x}{36 b \,p^{2} x^{3}+96 b p \,x^{3}+55 b \,x^{3}+36 a \,p^{2}+96 a p +55 a}d x \right ) a b c \,p^{2}+1815 \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p} x}{36 b \,p^{2} x^{3}+96 b p \,x^{3}+55 b \,x^{3}+36 a \,p^{2}+96 a p +55 a}d x \right ) a b c p \right )}{b \left (36 p^{2}+96 p +55\right )} \] Input:

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

Output:

(2*sqrt(e)*e*(6*sqrt(x)*(a + b*x**3)**p*a*d*p*x**2 + 6*sqrt(x)*(a + b*x**3 
)**p*b*c*p*x**2 + 11*sqrt(x)*(a + b*x**3)**p*b*c*x**2 + 6*sqrt(x)*(a + b*x 
**3)**p*b*d*p*x**5 + 5*sqrt(x)*(a + b*x**3)**p*b*d*x**5 - 540*int((sqrt(x) 
*(a + b*x**3)**p*x)/(36*a*p**2 + 96*a*p + 55*a + 36*b*p**2*x**3 + 96*b*p*x 
**3 + 55*b*x**3),x)*a**2*d*p**3 - 1440*int((sqrt(x)*(a + b*x**3)**p*x)/(36 
*a*p**2 + 96*a*p + 55*a + 36*b*p**2*x**3 + 96*b*p*x**3 + 55*b*x**3),x)*a** 
2*d*p**2 - 825*int((sqrt(x)*(a + b*x**3)**p*x)/(36*a*p**2 + 96*a*p + 55*a 
+ 36*b*p**2*x**3 + 96*b*p*x**3 + 55*b*x**3),x)*a**2*d*p + 648*int((sqrt(x) 
*(a + b*x**3)**p*x)/(36*a*p**2 + 96*a*p + 55*a + 36*b*p**2*x**3 + 96*b*p*x 
**3 + 55*b*x**3),x)*a*b*c*p**4 + 2916*int((sqrt(x)*(a + b*x**3)**p*x)/(36* 
a*p**2 + 96*a*p + 55*a + 36*b*p**2*x**3 + 96*b*p*x**3 + 55*b*x**3),x)*a*b* 
c*p**3 + 4158*int((sqrt(x)*(a + b*x**3)**p*x)/(36*a*p**2 + 96*a*p + 55*a + 
 36*b*p**2*x**3 + 96*b*p*x**3 + 55*b*x**3),x)*a*b*c*p**2 + 1815*int((sqrt( 
x)*(a + b*x**3)**p*x)/(36*a*p**2 + 96*a*p + 55*a + 36*b*p**2*x**3 + 96*b*p 
*x**3 + 55*b*x**3),x)*a*b*c*p))/(b*(36*p**2 + 96*p + 55))