Integrand size = 22, antiderivative size = 177 \[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^2 \, dx=\frac {2}{585} \left (117 c^2+\frac {4 a d (13 b c+3 a d)}{b^2}\right ) x \left (a-b x^2\right )^{3/4}-\frac {4 d (13 b c+3 a d) x \left (a-b x^2\right )^{7/4}}{117 b^2}-\frac {2 d^2 x^3 \left (a-b x^2\right )^{7/4}}{13 b}+\frac {2 a^{3/2} \left (117 b^2 c^2+4 a d (13 b c+3 a d)\right ) \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{195 b^{5/2} \sqrt [4]{a-b x^2}} \] Output:
2/585*(117*c^2+4*a*d*(3*a*d+13*b*c)/b^2)*x*(-b*x^2+a)^(3/4)-4/117*d*(3*a*d +13*b*c)*x*(-b*x^2+a)^(7/4)/b^2-2/13*d^2*x^3*(-b*x^2+a)^(7/4)/b+2/195*a^(3 /2)*(117*b^2*c^2+4*a*d*(3*a*d+13*b*c))*(1-b*x^2/a)^(1/4)*EllipticE(sin(1/2 *arcsin(b^(1/2)*x/a^(1/2))),2^(1/2))/b^(5/2)/(-b*x^2+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.90 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00 \[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^2 \, dx=\frac {x \left (a-b x^2\right )^{3/4} \left (7 a \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \operatorname {Gamma}\left (-\frac {3}{4}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {7}{2},\frac {b x^2}{a}\right )+8 b x^2 \left (2 c^2+3 c d x^2+d^2 x^4\right ) \operatorname {Gamma}\left (\frac {1}{4}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{2},\frac {9}{2},\frac {b x^2}{a}\right )+4 b x^2 \left (c+d x^2\right )^2 \operatorname {Gamma}\left (\frac {1}{4}\right ) \, _3F_2\left (\frac {1}{4},\frac {3}{2},2;1,\frac {9}{2};\frac {b x^2}{a}\right )\right )}{105 a \left (1-\frac {b x^2}{a}\right )^{3/4} \operatorname {Gamma}\left (-\frac {3}{4}\right )} \] Input:
Integrate[(a - b*x^2)^(3/4)*(c + d*x^2)^2,x]
Output:
(x*(a - b*x^2)^(3/4)*(7*a*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*Gamma[-3/4]*Hy pergeometric2F1[-3/4, 1/2, 7/2, (b*x^2)/a] + 8*b*x^2*(2*c^2 + 3*c*d*x^2 + d^2*x^4)*Gamma[1/4]*Hypergeometric2F1[1/4, 3/2, 9/2, (b*x^2)/a] + 4*b*x^2* (c + d*x^2)^2*Gamma[1/4]*HypergeometricPFQ[{1/4, 3/2, 2}, {1, 9/2}, (b*x^2 )/a]))/(105*a*(1 - (b*x^2)/a)^(3/4)*Gamma[-3/4])
Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {318, 27, 299, 211, 227, 226}
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.
| \(\displaystyle \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle -\frac {2 \int -\frac {1}{2} \left (a-b x^2\right )^{3/4} \left (d (17 b c+6 a d) x^2+c (13 b c+2 a d)\right )dx}{13 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )}{13 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \left (a-b x^2\right )^{3/4} \left (d (17 b c+6 a d) x^2+c (13 b c+2 a d)\right )dx}{13 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )}{13 b}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {\left (12 a^2 d^2+52 a b c d+117 b^2 c^2\right ) \int \left (a-b x^2\right )^{3/4}dx}{9 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} (6 a d+17 b c)}{9 b}}{13 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )}{13 b}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {\left (12 a^2 d^2+52 a b c d+117 b^2 c^2\right ) \left (\frac {3}{5} a \int \frac {1}{\sqrt [4]{a-b x^2}}dx+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{9 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} (6 a d+17 b c)}{9 b}}{13 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )}{13 b}\) |
| \(\Big \downarrow \) 227 |
| \(\displaystyle \frac {\frac {\left (12 a^2 d^2+52 a b c d+117 b^2 c^2\right ) \left (\frac {3 a \sqrt [4]{1-\frac {b x^2}{a}} \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}}dx}{5 \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{9 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} (6 a d+17 b c)}{9 b}}{13 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )}{13 b}\) |
| \(\Big \downarrow \) 226 |
| \(\displaystyle \frac {\frac {\left (12 a^2 d^2+52 a b c d+117 b^2 c^2\right ) \left (\frac {6 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{9 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} (6 a d+17 b c)}{9 b}}{13 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )}{13 b}\) |
Input:
Int[(a - b*x^2)^(3/4)*(c + d*x^2)^2,x]
Output:
(-2*d*x*(a - b*x^2)^(7/4)*(c + d*x^2))/(13*b) + ((-2*d*(17*b*c + 6*a*d)*x* (a - b*x^2)^(7/4))/(9*b) + ((117*b^2*c^2 + 52*a*b*c*d + 12*a^2*d^2)*((2*x* (a - b*x^2)^(3/4))/5 + (6*a^(3/2)*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[( Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*Sqrt[b]*(a - b*x^2)^(1/4))))/(9*b))/(13*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2] ))*EllipticE[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( a + b*x^2)^(1/4) Int[1/(1 + b*(x^2/a))^(1/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
\[\int \left (-b \,x^{2}+a \right )^{\frac {3}{4}} \left (x^{2} d +c \right )^{2}d x\]
Input:
int((-b*x^2+a)^(3/4)*(d*x^2+c)^2,x)
Output:
int((-b*x^2+a)^(3/4)*(d*x^2+c)^2,x)
\[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^2 \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{2} \,d x } \] Input:
integrate((-b*x^2+a)^(3/4)*(d*x^2+c)^2,x, algorithm="fricas")
Output:
integral((d^2*x^4 + 2*c*d*x^2 + c^2)*(-b*x^2 + a)^(3/4), x)
Result contains complex when optimal does not.
Time = 1.48 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.59 \[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^2 \, dx=a^{\frac {3}{4}} c^{2} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + \frac {2 a^{\frac {3}{4}} c d x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{3} + \frac {a^{\frac {3}{4}} d^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} \] Input:
integrate((-b*x**2+a)**(3/4)*(d*x**2+c)**2,x)
Output:
a**(3/4)*c**2*x*hyper((-3/4, 1/2), (3/2,), b*x**2*exp_polar(2*I*pi)/a) + 2 *a**(3/4)*c*d*x**3*hyper((-3/4, 3/2), (5/2,), b*x**2*exp_polar(2*I*pi)/a)/ 3 + a**(3/4)*d**2*x**5*hyper((-3/4, 5/2), (7/2,), b*x**2*exp_polar(2*I*pi) /a)/5
\[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^2 \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{2} \,d x } \] Input:
integrate((-b*x^2+a)^(3/4)*(d*x^2+c)^2,x, algorithm="maxima")
Output:
integrate((-b*x^2 + a)^(3/4)*(d*x^2 + c)^2, x)
\[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^2 \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{2} \,d x } \] Input:
integrate((-b*x^2+a)^(3/4)*(d*x^2+c)^2,x, algorithm="giac")
Output:
integrate((-b*x^2 + a)^(3/4)*(d*x^2 + c)^2, x)
Timed out. \[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^2 \, dx=\int {\left (a-b\,x^2\right )}^{3/4}\,{\left (d\,x^2+c\right )}^2 \,d x \] Input:
int((a - b*x^2)^(3/4)*(c + d*x^2)^2,x)
Output:
int((a - b*x^2)^(3/4)*(c + d*x^2)^2, x)
\[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^2 \, dx=\frac {-36 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a^{2} d^{2} x -156 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a b c d x -30 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a b \,d^{2} x^{3}+234 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} b^{2} c^{2} x +260 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} b^{2} c d \,x^{3}+90 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} b^{2} d^{2} x^{5}+36 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a^{3} d^{2}+156 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a^{2} b c d +351 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a \,b^{2} c^{2}}{585 b^{2}} \] Input:
int((-b*x^2+a)^(3/4)*(d*x^2+c)^2,x)
Output:
( - 36*(a - b*x**2)**(3/4)*a**2*d**2*x - 156*(a - b*x**2)**(3/4)*a*b*c*d*x - 30*(a - b*x**2)**(3/4)*a*b*d**2*x**3 + 234*(a - b*x**2)**(3/4)*b**2*c** 2*x + 260*(a - b*x**2)**(3/4)*b**2*c*d*x**3 + 90*(a - b*x**2)**(3/4)*b**2* d**2*x**5 + 36*int((a - b*x**2)**(3/4)/(a - b*x**2),x)*a**3*d**2 + 156*int ((a - b*x**2)**(3/4)/(a - b*x**2),x)*a**2*b*c*d + 351*int((a - b*x**2)**(3 /4)/(a - b*x**2),x)*a*b**2*c**2)/(585*b**2)