Integrand size = 24, antiderivative size = 102 \[ \int \frac {\left (a+b x^3\right )^p \left (c+d x^3\right )}{(e x)^{7/2}} \, dx=-\frac {2 c \left (a+b x^3\right )^{1+p}}{5 a e (e x)^{5/2}}+\frac {2 (5 a d+b (c+6 c p)) \sqrt {e x} \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},-p,\frac {7}{6},-\frac {b x^3}{a}\right )}{5 a e^4} \] Output:
-2/5*c*(b*x^3+a)^(p+1)/a/e/(e*x)^(5/2)+2/5*(5*a*d+b*(6*c*p+c))*(e*x)^(1/2) *(b*x^3+a)^p*hypergeom([1/6, -p],[7/6],-b*x^3/a)/a/e^4/((1+b*x^3/a)^p)
Time = 0.58 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^3\right )^p \left (c+d x^3\right )}{(e x)^{7/2}} \, dx=-\frac {2 x \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \left (c \left (a+b x^3\right ) \left (1+\frac {b x^3}{a}\right )^p-(5 a d+b (c+6 c p)) x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{6},-p,\frac {7}{6},-\frac {b x^3}{a}\right )\right )}{5 a (e x)^{7/2}} \] Input:
Integrate[((a + b*x^3)^p*(c + d*x^3))/(e*x)^(7/2),x]
Output:
(-2*x*(a + b*x^3)^p*(c*(a + b*x^3)*(1 + (b*x^3)/a)^p - (5*a*d + b*(c + 6*c *p))*x^3*Hypergeometric2F1[1/6, -p, 7/6, -((b*x^3)/a)]))/(5*a*(e*x)^(7/2)* (1 + (b*x^3)/a)^p)
Time = 0.44 (sec) , antiderivative size = 102, 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 = {955, 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.
| \(\displaystyle \int \frac {\left (c+d x^3\right ) \left (a+b x^3\right )^p}{(e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle \frac {(5 a d+b (6 c p+c)) \int \frac {\left (b x^3+a\right )^p}{\sqrt {e x}}dx}{5 a e^3}-\frac {2 c \left (a+b x^3\right )^{p+1}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 889 |
| \(\displaystyle \frac {\left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} (5 a d+b (6 c p+c)) \int \frac {\left (\frac {b x^3}{a}+1\right )^p}{\sqrt {e x}}dx}{5 a e^3}-\frac {2 c \left (a+b x^3\right )^{p+1}}{5 a e (e x)^{5/2}}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \frac {2 \sqrt {e x} \left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} (5 a d+b (6 c p+c)) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},-p,\frac {7}{6},-\frac {b x^3}{a}\right )}{5 a e^4}-\frac {2 c \left (a+b x^3\right )^{p+1}}{5 a e (e x)^{5/2}}\) |
Input:
Int[((a + b*x^3)^p*(c + d*x^3))/(e*x)^(7/2),x]
Output:
(-2*c*(a + b*x^3)^(1 + p))/(5*a*e*(e*x)^(5/2)) + (2*(5*a*d + b*(c + 6*c*p) )*Sqrt[e*x]*(a + b*x^3)^p*Hypergeometric2F1[1/6, -p, 7/6, -((b*x^3)/a)])/( 5*a*e^4*(1 + (b*x^3)/a)^p)
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])
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])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
\[\int \frac {\left (b \,x^{3}+a \right )^{p} \left (d \,x^{3}+c \right )}{\left (e x \right )^{\frac {7}{2}}}d x\]
Input:
int((b*x^3+a)^p*(d*x^3+c)/(e*x)^(7/2),x)
Output:
int((b*x^3+a)^p*(d*x^3+c)/(e*x)^(7/2),x)
\[ \int \frac {\left (a+b x^3\right )^p \left (c+d x^3\right )}{(e x)^{7/2}} \, dx=\int { \frac {{\left (d x^{3} + c\right )} {\left (b x^{3} + a\right )}^{p}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((b*x^3+a)^p*(d*x^3+c)/(e*x)^(7/2),x, algorithm="fricas")
Output:
integral((d*x^3 + c)*sqrt(e*x)*(b*x^3 + a)^p/(e^4*x^4), x)
Timed out. \[ \int \frac {\left (a+b x^3\right )^p \left (c+d x^3\right )}{(e x)^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x**3+a)**p*(d*x**3+c)/(e*x)**(7/2),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^3\right )^p \left (c+d x^3\right )}{(e x)^{7/2}} \, dx=\int { \frac {{\left (d x^{3} + c\right )} {\left (b x^{3} + a\right )}^{p}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((b*x^3+a)^p*(d*x^3+c)/(e*x)^(7/2),x, algorithm="maxima")
Output:
integrate((d*x^3 + c)*(b*x^3 + a)^p/(e*x)^(7/2), x)
\[ \int \frac {\left (a+b x^3\right )^p \left (c+d x^3\right )}{(e x)^{7/2}} \, dx=\int { \frac {{\left (d x^{3} + c\right )} {\left (b x^{3} + a\right )}^{p}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((b*x^3+a)^p*(d*x^3+c)/(e*x)^(7/2),x, algorithm="giac")
Output:
integrate((d*x^3 + c)*(b*x^3 + a)^p/(e*x)^(7/2), x)
Timed out. \[ \int \frac {\left (a+b x^3\right )^p \left (c+d x^3\right )}{(e x)^{7/2}} \, dx=\int \frac {{\left (b\,x^3+a\right )}^p\,\left (d\,x^3+c\right )}{{\left (e\,x\right )}^{7/2}} \,d x \] Input:
int(((a + b*x^3)^p*(c + d*x^3))/(e*x)^(7/2),x)
Output:
int(((a + b*x^3)^p*(c + d*x^3))/(e*x)^(7/2), x)
\[ \int \frac {\left (a+b x^3\right )^p \left (c+d x^3\right )}{(e x)^{7/2}} \, dx=\frac {2 \sqrt {e}\, \left (6 \left (b \,x^{3}+a \right )^{p} a d p +6 \left (b \,x^{3}+a \right )^{p} b c p +\left (b \,x^{3}+a \right )^{p} b c +6 \left (b \,x^{3}+a \right )^{p} b d p \,x^{3}-5 \left (b \,x^{3}+a \right )^{p} b d \,x^{3}+540 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p}}{36 b \,p^{2} x^{7}-24 b p \,x^{7}-5 b \,x^{7}+36 a \,p^{2} x^{4}-24 a p \,x^{4}-5 a \,x^{4}}d x \right ) a^{2} d \,p^{3} x^{2}-360 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p}}{36 b \,p^{2} x^{7}-24 b p \,x^{7}-5 b \,x^{7}+36 a \,p^{2} x^{4}-24 a p \,x^{4}-5 a \,x^{4}}d x \right ) a^{2} d \,p^{2} x^{2}-75 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p}}{36 b \,p^{2} x^{7}-24 b p \,x^{7}-5 b \,x^{7}+36 a \,p^{2} x^{4}-24 a p \,x^{4}-5 a \,x^{4}}d x \right ) a^{2} d p \,x^{2}+648 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p}}{36 b \,p^{2} x^{7}-24 b p \,x^{7}-5 b \,x^{7}+36 a \,p^{2} x^{4}-24 a p \,x^{4}-5 a \,x^{4}}d x \right ) a b c \,p^{4} x^{2}-324 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p}}{36 b \,p^{2} x^{7}-24 b p \,x^{7}-5 b \,x^{7}+36 a \,p^{2} x^{4}-24 a p \,x^{4}-5 a \,x^{4}}d x \right ) a b c \,p^{3} x^{2}-162 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p}}{36 b \,p^{2} x^{7}-24 b p \,x^{7}-5 b \,x^{7}+36 a \,p^{2} x^{4}-24 a p \,x^{4}-5 a \,x^{4}}d x \right ) a b c \,p^{2} x^{2}-15 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{3}+a \right )^{p}}{36 b \,p^{2} x^{7}-24 b p \,x^{7}-5 b \,x^{7}+36 a \,p^{2} x^{4}-24 a p \,x^{4}-5 a \,x^{4}}d x \right ) a b c p \,x^{2}\right )}{\sqrt {x}\, b \,e^{4} x^{2} \left (36 p^{2}-24 p -5\right )} \] Input:
int((b*x^3+a)^p*(d*x^3+c)/(e*x)^(7/2),x)
Output:
(2*sqrt(e)*(6*(a + b*x**3)**p*a*d*p + 6*(a + b*x**3)**p*b*c*p + (a + b*x** 3)**p*b*c + 6*(a + b*x**3)**p*b*d*p*x**3 - 5*(a + b*x**3)**p*b*d*x**3 + 54 0*sqrt(x)*int((sqrt(x)*(a + b*x**3)**p)/(36*a*p**2*x**4 - 24*a*p*x**4 - 5* a*x**4 + 36*b*p**2*x**7 - 24*b*p*x**7 - 5*b*x**7),x)*a**2*d*p**3*x**2 - 36 0*sqrt(x)*int((sqrt(x)*(a + b*x**3)**p)/(36*a*p**2*x**4 - 24*a*p*x**4 - 5* a*x**4 + 36*b*p**2*x**7 - 24*b*p*x**7 - 5*b*x**7),x)*a**2*d*p**2*x**2 - 75 *sqrt(x)*int((sqrt(x)*(a + b*x**3)**p)/(36*a*p**2*x**4 - 24*a*p*x**4 - 5*a *x**4 + 36*b*p**2*x**7 - 24*b*p*x**7 - 5*b*x**7),x)*a**2*d*p*x**2 + 648*sq rt(x)*int((sqrt(x)*(a + b*x**3)**p)/(36*a*p**2*x**4 - 24*a*p*x**4 - 5*a*x* *4 + 36*b*p**2*x**7 - 24*b*p*x**7 - 5*b*x**7),x)*a*b*c*p**4*x**2 - 324*sqr t(x)*int((sqrt(x)*(a + b*x**3)**p)/(36*a*p**2*x**4 - 24*a*p*x**4 - 5*a*x** 4 + 36*b*p**2*x**7 - 24*b*p*x**7 - 5*b*x**7),x)*a*b*c*p**3*x**2 - 162*sqrt (x)*int((sqrt(x)*(a + b*x**3)**p)/(36*a*p**2*x**4 - 24*a*p*x**4 - 5*a*x**4 + 36*b*p**2*x**7 - 24*b*p*x**7 - 5*b*x**7),x)*a*b*c*p**2*x**2 - 15*sqrt(x )*int((sqrt(x)*(a + b*x**3)**p)/(36*a*p**2*x**4 - 24*a*p*x**4 - 5*a*x**4 + 36*b*p**2*x**7 - 24*b*p*x**7 - 5*b*x**7),x)*a*b*c*p*x**2))/(sqrt(x)*b*e** 4*x**2*(36*p**2 - 24*p - 5))