Integrand size = 22, antiderivative size = 115 \[ \int (e x)^{3/2} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {2 c (e x)^{5/2} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^2}{a}\right )}{5 e}+\frac {2 d (e x)^{7/2} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{4},-p,\frac {11}{4},-\frac {b x^2}{a}\right )}{7 e^2} \] Output:
2/5*c*(e*x)^(5/2)*(b*x^2+a)^p*hypergeom([5/4, -p],[9/4],-b*x^2/a)/e/((1+b* x^2/a)^p)+2/7*d*(e*x)^(7/2)*(b*x^2+a)^p*hypergeom([7/4, -p],[11/4],-b*x^2/ a)/e^2/((1+b*x^2/a)^p)
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.70 \[ \int (e x)^{3/2} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {2}{35} x (e x)^{3/2} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (7 c \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^2}{a}\right )+5 d x \operatorname {Hypergeometric2F1}\left (\frac {7}{4},-p,\frac {11}{4},-\frac {b x^2}{a}\right )\right ) \] Input:
Integrate[(e*x)^(3/2)*(c + d*x)*(a + b*x^2)^p,x]
Output:
(2*x*(e*x)^(3/2)*(a + b*x^2)^p*(7*c*Hypergeometric2F1[5/4, -p, 9/4, -((b*x ^2)/a)] + 5*d*x*Hypergeometric2F1[7/4, -p, 11/4, -((b*x^2)/a)]))/(35*(1 + (b*x^2)/a)^p)
Time = 0.35 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {557, 279, 278}
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.
\(\displaystyle \int (e x)^{3/2} (c+d x) \left (a+b x^2\right )^p \, dx\) |
\(\Big \downarrow \) 557 |
\(\displaystyle c \int (e x)^{3/2} \left (b x^2+a\right )^pdx+\frac {d \int (e x)^{5/2} \left (b x^2+a\right )^pdx}{e}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle c \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^{3/2} \left (\frac {b x^2}{a}+1\right )^pdx+\frac {d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^{5/2} \left (\frac {b x^2}{a}+1\right )^pdx}{e}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {2 c (e x)^{5/2} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^2}{a}\right )}{5 e}+\frac {2 d (e x)^{7/2} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{4},-p,\frac {11}{4},-\frac {b x^2}{a}\right )}{7 e^2}\) |
Input:
Int[(e*x)^(3/2)*(c + d*x)*(a + b*x^2)^p,x]
Output:
(2*c*(e*x)^(5/2)*(a + b*x^2)^p*Hypergeometric2F1[5/4, -p, 9/4, -((b*x^2)/a )])/(5*e*(1 + (b*x^2)/a)^p) + (2*d*(e*x)^(7/2)*(a + b*x^2)^p*Hypergeometri c2F1[7/4, -p, 11/4, -((b*x^2)/a)])/(7*e^2*(1 + (b*x^2)/a)^p)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(c*x)^m* (1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
\[\int \left (e x \right )^{\frac {3}{2}} \left (d x +c \right ) \left (b \,x^{2}+a \right )^{p}d x\]
Input:
int((e*x)^(3/2)*(d*x+c)*(b*x^2+a)^p,x)
Output:
int((e*x)^(3/2)*(d*x+c)*(b*x^2+a)^p,x)
\[ \int (e x)^{3/2} (c+d x) \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )} \left (e x\right )^{\frac {3}{2}} {\left (b x^{2} + a\right )}^{p} \,d x } \] Input:
integrate((e*x)^(3/2)*(d*x+c)*(b*x^2+a)^p,x, algorithm="fricas")
Output:
integral((d*e*x^2 + c*e*x)*sqrt(e*x)*(b*x^2 + a)^p, x)
Result contains complex when optimal does not.
Time = 154.72 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.78 \[ \int (e x)^{3/2} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {a^{p} c e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, - p \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {a^{p} d e^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, - p \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((e*x)**(3/2)*(d*x+c)*(b*x**2+a)**p,x)
Output:
a**p*c*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((5/4, -p), (9/4,), b*x**2*exp_po lar(I*pi)/a)/(2*gamma(9/4)) + a**p*d*e**(3/2)*x**(7/2)*gamma(7/4)*hyper((7 /4, -p), (11/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(11/4))
\[ \int (e x)^{3/2} (c+d x) \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )} \left (e x\right )^{\frac {3}{2}} {\left (b x^{2} + a\right )}^{p} \,d x } \] Input:
integrate((e*x)^(3/2)*(d*x+c)*(b*x^2+a)^p,x, algorithm="maxima")
Output:
integrate((d*x + c)*(e*x)^(3/2)*(b*x^2 + a)^p, x)
\[ \int (e x)^{3/2} (c+d x) \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )} \left (e x\right )^{\frac {3}{2}} {\left (b x^{2} + a\right )}^{p} \,d x } \] Input:
integrate((e*x)^(3/2)*(d*x+c)*(b*x^2+a)^p,x, algorithm="giac")
Output:
integrate((d*x + c)*(e*x)^(3/2)*(b*x^2 + a)^p, x)
Timed out. \[ \int (e x)^{3/2} (c+d x) \left (a+b x^2\right )^p \, dx=\int {\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^p\,\left (c+d\,x\right ) \,d x \] Input:
int((e*x)^(3/2)*(a + b*x^2)^p*(c + d*x),x)
Output:
int((e*x)^(3/2)*(a + b*x^2)^p*(c + d*x), x)
\[ \int (e x)^{3/2} (c+d x) \left (a+b x^2\right )^p \, dx=\text {too large to display} \] Input:
int((e*x)^(3/2)*(d*x+c)*(b*x^2+a)^p,x)
Output:
(2*sqrt(e)*e*(64*sqrt(x)*(a + b*x**2)**p*a*c*p**3 + 160*sqrt(x)*(a + b*x** 2)**p*a*c*p**2 + 84*sqrt(x)*(a + b*x**2)**p*a*c*p + 64*sqrt(x)*(a + b*x**2 )**p*a*d*p**3*x + 96*sqrt(x)*(a + b*x**2)**p*a*d*p**2*x + 20*sqrt(x)*(a + b*x**2)**p*a*d*p*x + 64*sqrt(x)*(a + b*x**2)**p*b*c*p**3*x**2 + 176*sqrt(x )*(a + b*x**2)**p*b*c*p**2*x**2 + 124*sqrt(x)*(a + b*x**2)**p*b*c*p*x**2 + 21*sqrt(x)*(a + b*x**2)**p*b*c*x**2 + 64*sqrt(x)*(a + b*x**2)**p*b*d*p**3 *x**3 + 144*sqrt(x)*(a + b*x**2)**p*b*d*p**2*x**3 + 92*sqrt(x)*(a + b*x**2 )**p*b*d*p*x**3 + 15*sqrt(x)*(a + b*x**2)**p*b*d*x**3 - 8192*int((sqrt(x)* (a + b*x**2)**p)/(256*a*p**4*x + 1024*a*p**3*x + 1376*a*p**2*x + 704*a*p*x + 105*a*x + 256*b*p**4*x**3 + 1024*b*p**3*x**3 + 1376*b*p**2*x**3 + 704*b *p*x**3 + 105*b*x**3),x)*a**2*c*p**7 - 53248*int((sqrt(x)*(a + b*x**2)**p) /(256*a*p**4*x + 1024*a*p**3*x + 1376*a*p**2*x + 704*a*p*x + 105*a*x + 256 *b*p**4*x**3 + 1024*b*p**3*x**3 + 1376*b*p**2*x**3 + 704*b*p*x**3 + 105*b* x**3),x)*a**2*c*p**6 - 136704*int((sqrt(x)*(a + b*x**2)**p)/(256*a*p**4*x + 1024*a*p**3*x + 1376*a*p**2*x + 704*a*p*x + 105*a*x + 256*b*p**4*x**3 + 1024*b*p**3*x**3 + 1376*b*p**2*x**3 + 704*b*p*x**3 + 105*b*x**3),x)*a**2*c *p**5 - 175616*int((sqrt(x)*(a + b*x**2)**p)/(256*a*p**4*x + 1024*a*p**3*x + 1376*a*p**2*x + 704*a*p*x + 105*a*x + 256*b*p**4*x**3 + 1024*b*p**3*x** 3 + 1376*b*p**2*x**3 + 704*b*p*x**3 + 105*b*x**3),x)*a**2*c*p**4 - 117472* int((sqrt(x)*(a + b*x**2)**p)/(256*a*p**4*x + 1024*a*p**3*x + 1376*a*p*...