\(\int (a-b x^2)^{3/4} (c+d x^2)^3 \, dx\) [455]

Optimal result

Integrand size = 22, antiderivative size = 253 \[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^3 \, dx=\frac {2 \left (663 c^3+\frac {2 a d \left (221 b^2 c^2+102 a b c d+20 a^2 d^2\right )}{b^3}\right ) x \left (a-b x^2\right )^{3/4}}{3315}-\frac {2 d \left (221 b^2 c^2+102 a b c d+20 a^2 d^2\right ) x \left (a-b x^2\right )^{7/4}}{663 b^3}-\frac {2 d^2 (51 b c+10 a d) x^3 \left (a-b x^2\right )^{7/4}}{221 b^2}-\frac {2 d^3 x^5 \left (a-b x^2\right )^{7/4}}{17 b}+\frac {2 a^{3/2} \left (663 b^3 c^3+2 a d \left (221 b^2 c^2+102 a b c d+20 a^2 d^2\right )\right ) \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{1105 b^{7/2} \sqrt [4]{a-b x^2}} \] Output:

2/3315*(663*c^3+2*a*d*(20*a^2*d^2+102*a*b*c*d+221*b^2*c^2)/b^3)*x*(-b*x^2+ 
a)^(3/4)-2/663*d*(20*a^2*d^2+102*a*b*c*d+221*b^2*c^2)*x*(-b*x^2+a)^(7/4)/b 
^3-2/221*d^2*(10*a*d+51*b*c)*x^3*(-b*x^2+a)^(7/4)/b^2-2/17*d^3*x^5*(-b*x^2 
+a)^(7/4)/b+2/1105*a^(3/2)*(663*b^3*c^3+2*a*d*(20*a^2*d^2+102*a*b*c*d+221* 
b^2*c^2))*(1-b*x^2/a)^(1/4)*EllipticE(sin(1/2*arcsin(b^(1/2)*x/a^(1/2))),2 
^(1/2))/b^(7/2)/(-b*x^2+a)^(1/4)
 

Mathematica [C] (verified)

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

Time = 13.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.78 \[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^3 \, dx=\frac {x \left (-2 \left (a-b x^2\right ) \left (60 a^3 d^3+2 a^2 b d^2 \left (153 c+25 d x^2\right )+3 a b^2 d \left (221 c^2+85 c d x^2+15 d^2 x^4\right )-b^3 \left (663 c^3+1105 c^2 d x^2+765 c d^2 x^4+195 d^3 x^6\right )\right )+3 a \left (663 b^3 c^3+442 a b^2 c^2 d+204 a^2 b c d^2+40 a^3 d^3\right ) \sqrt [4]{1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {b x^2}{a}\right )\right )}{3315 b^3 \sqrt [4]{a-b x^2}} \] Input:

Integrate[(a - b*x^2)^(3/4)*(c + d*x^2)^3,x]
 

Output:

(x*(-2*(a - b*x^2)*(60*a^3*d^3 + 2*a^2*b*d^2*(153*c + 25*d*x^2) + 3*a*b^2* 
d*(221*c^2 + 85*c*d*x^2 + 15*d^2*x^4) - b^3*(663*c^3 + 1105*c^2*d*x^2 + 76 
5*c*d^2*x^4 + 195*d^3*x^6)) + 3*a*(663*b^3*c^3 + 442*a*b^2*c^2*d + 204*a^2 
*b*c*d^2 + 40*a^3*d^3)*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3 
/2, (b*x^2)/a]))/(3315*b^3*(a - b*x^2)^(1/4))
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {318, 27, 403, 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.

\(\displaystyle \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^3 \, dx\)

\(\Big \downarrow \) 318

\(\displaystyle -\frac {2 \int -\frac {1}{2} \left (a-b x^2\right )^{3/4} \left (d x^2+c\right ) \left (5 d (5 b c+2 a d) x^2+c (17 b c+2 a d)\right )dx}{17 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )^2}{17 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \left (a-b x^2\right )^{3/4} \left (d x^2+c\right ) \left (5 d (5 b c+2 a d) x^2+c (17 b c+2 a d)\right )dx}{17 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )^2}{17 b}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {-\frac {2 \int -\frac {1}{2} \left (a-b x^2\right )^{3/4} \left (3 d \left (107 b^2 c^2+72 a b d c+20 a^2 d^2\right ) x^2+c \left (221 b^2 c^2+76 a b d c+20 a^2 d^2\right )\right )dx}{13 b}-\frac {10 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right ) (2 a d+5 b c)}{13 b}}{17 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )^2}{17 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \left (a-b x^2\right )^{3/4} \left (3 d \left (107 b^2 c^2+72 a b d c+20 a^2 d^2\right ) x^2+c \left (221 b^2 c^2+76 a b d c+20 a^2 d^2\right )\right )dx}{13 b}-\frac {10 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right ) (2 a d+5 b c)}{13 b}}{17 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )^2}{17 b}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {\left (40 a^3 d^3+204 a^2 b c d^2+442 a b^2 c^2 d+663 b^3 c^3\right ) \int \left (a-b x^2\right )^{3/4}dx}{3 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (20 a^2 d^2+72 a b c d+107 b^2 c^2\right )}{3 b}}{13 b}-\frac {10 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right ) (2 a d+5 b c)}{13 b}}{17 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )^2}{17 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\left (40 a^3 d^3+204 a^2 b c d^2+442 a b^2 c^2 d+663 b^3 c^3\right ) \left (\frac {3}{5} a \int \frac {1}{\sqrt [4]{a-b x^2}}dx+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{3 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (20 a^2 d^2+72 a b c d+107 b^2 c^2\right )}{3 b}}{13 b}-\frac {10 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right ) (2 a d+5 b c)}{13 b}}{17 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )^2}{17 b}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {\frac {\frac {\left (40 a^3 d^3+204 a^2 b c d^2+442 a b^2 c^2 d+663 b^3 c^3\right ) \left (\frac {3 a \sqrt [4]{1-\frac {b x^2}{a}} \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}}dx}{5 \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{3 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (20 a^2 d^2+72 a b c d+107 b^2 c^2\right )}{3 b}}{13 b}-\frac {10 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right ) (2 a d+5 b c)}{13 b}}{17 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )^2}{17 b}\)

\(\Big \downarrow \) 226

\(\displaystyle \frac {\frac {\frac {\left (40 a^3 d^3+204 a^2 b c d^2+442 a b^2 c^2 d+663 b^3 c^3\right ) \left (\frac {6 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{3 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (20 a^2 d^2+72 a b c d+107 b^2 c^2\right )}{3 b}}{13 b}-\frac {10 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right ) (2 a d+5 b c)}{13 b}}{17 b}-\frac {2 d x \left (a-b x^2\right )^{7/4} \left (c+d x^2\right )^2}{17 b}\)

Input:

Int[(a - b*x^2)^(3/4)*(c + d*x^2)^3,x]
 

Output:

(-2*d*x*(a - b*x^2)^(7/4)*(c + d*x^2)^2)/(17*b) + ((-10*d*(5*b*c + 2*a*d)* 
x*(a - b*x^2)^(7/4)*(c + d*x^2))/(13*b) + ((-2*d*(107*b^2*c^2 + 72*a*b*c*d 
 + 20*a^2*d^2)*x*(a - b*x^2)^(7/4))/(3*b) + ((663*b^3*c^3 + 442*a*b^2*c^2* 
d + 204*a^2*b*c*d^2 + 40*a^3*d^3)*((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))))/(3*b))/(13*b))/(17*b)
 

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 211

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])
 

rule 226

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]
 

rule 227

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]
 

rule 299

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]
 

rule 318

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]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

Maple [F]

\[\int \left (-b \,x^{2}+a \right )^{\frac {3}{4}} \left (x^{2} d +c \right )^{3}d x\]

Input:

int((-b*x^2+a)^(3/4)*(d*x^2+c)^3,x)
 

Output:

int((-b*x^2+a)^(3/4)*(d*x^2+c)^3,x)
 

Fricas [F]

\[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^3 \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{3} \,d x } \] Input:

integrate((-b*x^2+a)^(3/4)*(d*x^2+c)^3,x, algorithm="fricas")
 

Output:

integral((d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3)*(-b*x^2 + a)^(3/4), x 
)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.04 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.56 \[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^3 \, dx=a^{\frac {3}{4}} c^{3} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + a^{\frac {3}{4}} c^{2} d x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + \frac {3 a^{\frac {3}{4}} c d^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} + \frac {a^{\frac {3}{4}} d^{3} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{7} \] Input:

integrate((-b*x**2+a)**(3/4)*(d*x**2+c)**3,x)
 

Output:

a**(3/4)*c**3*x*hyper((-3/4, 1/2), (3/2,), b*x**2*exp_polar(2*I*pi)/a) + a 
**(3/4)*c**2*d*x**3*hyper((-3/4, 3/2), (5/2,), b*x**2*exp_polar(2*I*pi)/a) 
 + 3*a**(3/4)*c*d**2*x**5*hyper((-3/4, 5/2), (7/2,), b*x**2*exp_polar(2*I* 
pi)/a)/5 + a**(3/4)*d**3*x**7*hyper((-3/4, 7/2), (9/2,), b*x**2*exp_polar( 
2*I*pi)/a)/7
 

Maxima [F]

\[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^3 \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{3} \,d x } \] Input:

integrate((-b*x^2+a)^(3/4)*(d*x^2+c)^3,x, algorithm="maxima")
 

Output:

integrate((-b*x^2 + a)^(3/4)*(d*x^2 + c)^3, x)
 

Giac [F]

\[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^3 \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{3} \,d x } \] Input:

integrate((-b*x^2+a)^(3/4)*(d*x^2+c)^3,x, algorithm="giac")
 

Output:

integrate((-b*x^2 + a)^(3/4)*(d*x^2 + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^3 \, dx=\int {\left (a-b\,x^2\right )}^{3/4}\,{\left (d\,x^2+c\right )}^3 \,d x \] Input:

int((a - b*x^2)^(3/4)*(c + d*x^2)^3,x)
 

Output:

int((a - b*x^2)^(3/4)*(c + d*x^2)^3, x)
 

Reduce [F]

\[ \int \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )^3 \, dx=\frac {-120 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a^{3} d^{3} x -612 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a^{2} b c \,d^{2} x -100 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a^{2} b \,d^{3} x^{3}-1326 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a \,b^{2} c^{2} d x -510 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a \,b^{2} c \,d^{2} x^{3}-90 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a \,b^{2} d^{3} x^{5}+1326 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} b^{3} c^{3} x +2210 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} b^{3} c^{2} d \,x^{3}+1530 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} b^{3} c \,d^{2} x^{5}+390 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} b^{3} d^{3} x^{7}+120 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a^{4} d^{3}+612 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a^{3} b c \,d^{2}+1326 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a^{2} b^{2} c^{2} d +1989 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a \,b^{3} c^{3}}{3315 b^{3}} \] Input:

int((-b*x^2+a)^(3/4)*(d*x^2+c)^3,x)
 

Output:

( - 120*(a - b*x**2)**(3/4)*a**3*d**3*x - 612*(a - b*x**2)**(3/4)*a**2*b*c 
*d**2*x - 100*(a - b*x**2)**(3/4)*a**2*b*d**3*x**3 - 1326*(a - b*x**2)**(3 
/4)*a*b**2*c**2*d*x - 510*(a - b*x**2)**(3/4)*a*b**2*c*d**2*x**3 - 90*(a - 
 b*x**2)**(3/4)*a*b**2*d**3*x**5 + 1326*(a - b*x**2)**(3/4)*b**3*c**3*x + 
2210*(a - b*x**2)**(3/4)*b**3*c**2*d*x**3 + 1530*(a - b*x**2)**(3/4)*b**3* 
c*d**2*x**5 + 390*(a - b*x**2)**(3/4)*b**3*d**3*x**7 + 120*int((a - b*x**2 
)**(3/4)/(a - b*x**2),x)*a**4*d**3 + 612*int((a - b*x**2)**(3/4)/(a - b*x* 
*2),x)*a**3*b*c*d**2 + 1326*int((a - b*x**2)**(3/4)/(a - b*x**2),x)*a**2*b 
**2*c**2*d + 1989*int((a - b*x**2)**(3/4)/(a - b*x**2),x)*a*b**3*c**3)/(33 
15*b**3)