\(\int (a-b x^2)^{5/4} (c+d x^2)^3 \, dx\) [460]

Optimal result

Integrand size = 22, antiderivative size = 310 \[ \int \left (a-b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=\frac {2 a \left (1045 b^3 c^3+2 a d \left (285 b^2 c^2+114 a b c d+20 a^2 d^2\right )\right ) x \sqrt [4]{a-b x^2}}{4389 b^3}+\frac {2 \left (1045 c^3+\frac {2 a d \left (285 b^2 c^2+114 a b c d+20 a^2 d^2\right )}{b^3}\right ) x \left (a-b x^2\right )^{5/4}}{7315}-\frac {2 d \left (285 b^2 c^2+114 a b c d+20 a^2 d^2\right ) x \left (a-b x^2\right )^{9/4}}{1045 b^3}-\frac {2 d^2 (57 b c+10 a d) x^3 \left (a-b x^2\right )^{9/4}}{285 b^2}-\frac {2 d^3 x^5 \left (a-b x^2\right )^{9/4}}{19 b}+\frac {2 a^{5/2} \left (1045 b^3 c^3+2 a d \left (285 b^2 c^2+114 a b c d+20 a^2 d^2\right )\right ) \left (1-\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{4389 b^{7/2} \left (a-b x^2\right )^{3/4}} \] Output:

2/4389*a*(1045*b^3*c^3+2*a*d*(20*a^2*d^2+114*a*b*c*d+285*b^2*c^2))*x*(-b*x 
^2+a)^(1/4)/b^3+2/7315*(1045*c^3+2*a*d*(20*a^2*d^2+114*a*b*c*d+285*b^2*c^2 
)/b^3)*x*(-b*x^2+a)^(5/4)-2/1045*d*(20*a^2*d^2+114*a*b*c*d+285*b^2*c^2)*x* 
(-b*x^2+a)^(9/4)/b^3-2/285*d^2*(10*a*d+57*b*c)*x^3*(-b*x^2+a)^(9/4)/b^2-2/ 
19*d^3*x^5*(-b*x^2+a)^(9/4)/b+2/4389*a^(5/2)*(1045*b^3*c^3+2*a*d*(20*a^2*d 
^2+114*a*b*c*d+285*b^2*c^2))*(1-b*x^2/a)^(3/4)*InverseJacobiAM(1/2*arcsin( 
b^(1/2)*x/a^(1/2)),2^(1/2))/b^(7/2)/(-b*x^2+a)^(3/4)
 

Mathematica [C] (verified)

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

Time = 13.99 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.78 \[ \int \left (a-b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=\frac {x \left (-2 \left (a-b x^2\right ) \left (100 a^4 d^3+10 a^3 b d^2 \left (57 c+5 d x^2\right )+5 a^2 b^2 d \left (285 c^2+57 c d x^2+7 d^2 x^4\right )+3 b^4 x^2 \left (1045 c^3+1995 c^2 d x^2+1463 c d^2 x^4+385 d^3 x^6\right )-4 a b^3 \left (2090 c^3+2565 c^2 d x^2+1596 c d^2 x^4+385 d^3 x^6\right )\right )+5 a^2 \left (1045 b^3 c^3+570 a b^2 c^2 d+228 a^2 b c d^2+40 a^3 d^3\right ) \left (1-\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {b x^2}{a}\right )\right )}{21945 b^3 \left (a-b x^2\right )^{3/4}} \] Input:

Integrate[(a - b*x^2)^(5/4)*(c + d*x^2)^3,x]
 

Output:

(x*(-2*(a - b*x^2)*(100*a^4*d^3 + 10*a^3*b*d^2*(57*c + 5*d*x^2) + 5*a^2*b^ 
2*d*(285*c^2 + 57*c*d*x^2 + 7*d^2*x^4) + 3*b^4*x^2*(1045*c^3 + 1995*c^2*d* 
x^2 + 1463*c*d^2*x^4 + 385*d^3*x^6) - 4*a*b^3*(2090*c^3 + 2565*c^2*d*x^2 + 
 1596*c*d^2*x^4 + 385*d^3*x^6)) + 5*a^2*(1045*b^3*c^3 + 570*a*b^2*c^2*d + 
228*a^2*b*c*d^2 + 40*a^3*d^3)*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 
 3/4, 3/2, (b*x^2)/a]))/(21945*b^3*(a - b*x^2)^(3/4))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.88, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {318, 27, 403, 27, 299, 211, 211, 231, 230}

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.

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

\(\Big \downarrow \) 318

\(\displaystyle -\frac {2 \int -\frac {1}{2} \left (a-b x^2\right )^{5/4} \left (d x^2+c\right ) \left (d (27 b c+10 a d) x^2+c (19 b c+2 a d)\right )dx}{19 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2}{19 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \left (a-b x^2\right )^{5/4} \left (d x^2+c\right ) \left (d (27 b c+10 a d) x^2+c (19 b c+2 a d)\right )dx}{19 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2}{19 b}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {-\frac {2 \int -\frac {1}{2} \left (a-b x^2\right )^{5/4} \left (d \left (393 b^2 c^2+232 a b d c+60 a^2 d^2\right ) x^2+c \left (285 b^2 c^2+84 a b d c+20 a^2 d^2\right )\right )dx}{15 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right ) (10 a d+27 b c)}{15 b}}{19 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2}{19 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \left (a-b x^2\right )^{5/4} \left (d \left (393 b^2 c^2+232 a b d c+60 a^2 d^2\right ) x^2+c \left (285 b^2 c^2+84 a b d c+20 a^2 d^2\right )\right )dx}{15 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right ) (10 a d+27 b c)}{15 b}}{19 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2}{19 b}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {3 \left (40 a^3 d^3+228 a^2 b c d^2+570 a b^2 c^2 d+1045 b^3 c^3\right ) \int \left (a-b x^2\right )^{5/4}dx}{11 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (60 a^2 d^2+232 a b c d+393 b^2 c^2\right )}{11 b}}{15 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right ) (10 a d+27 b c)}{15 b}}{19 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2}{19 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {3 \left (40 a^3 d^3+228 a^2 b c d^2+570 a b^2 c^2 d+1045 b^3 c^3\right ) \left (\frac {5}{7} a \int \sqrt [4]{a-b x^2}dx+\frac {2}{7} x \left (a-b x^2\right )^{5/4}\right )}{11 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (60 a^2 d^2+232 a b c d+393 b^2 c^2\right )}{11 b}}{15 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right ) (10 a d+27 b c)}{15 b}}{19 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2}{19 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {3 \left (40 a^3 d^3+228 a^2 b c d^2+570 a b^2 c^2 d+1045 b^3 c^3\right ) \left (\frac {5}{7} a \left (\frac {1}{3} a \int \frac {1}{\left (a-b x^2\right )^{3/4}}dx+\frac {2}{3} x \sqrt [4]{a-b x^2}\right )+\frac {2}{7} x \left (a-b x^2\right )^{5/4}\right )}{11 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (60 a^2 d^2+232 a b c d+393 b^2 c^2\right )}{11 b}}{15 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right ) (10 a d+27 b c)}{15 b}}{19 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2}{19 b}\)

\(\Big \downarrow \) 231

\(\displaystyle \frac {\frac {\frac {3 \left (40 a^3 d^3+228 a^2 b c d^2+570 a b^2 c^2 d+1045 b^3 c^3\right ) \left (\frac {5}{7} a \left (\frac {a \left (1-\frac {b x^2}{a}\right )^{3/4} \int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}}dx}{3 \left (a-b x^2\right )^{3/4}}+\frac {2}{3} x \sqrt [4]{a-b x^2}\right )+\frac {2}{7} x \left (a-b x^2\right )^{5/4}\right )}{11 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (60 a^2 d^2+232 a b c d+393 b^2 c^2\right )}{11 b}}{15 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right ) (10 a d+27 b c)}{15 b}}{19 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2}{19 b}\)

\(\Big \downarrow \) 230

\(\displaystyle \frac {\frac {\frac {3 \left (40 a^3 d^3+228 a^2 b c d^2+570 a b^2 c^2 d+1045 b^3 c^3\right ) \left (\frac {5}{7} a \left (\frac {2 a^{3/2} \left (1-\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 \sqrt {b} \left (a-b x^2\right )^{3/4}}+\frac {2}{3} x \sqrt [4]{a-b x^2}\right )+\frac {2}{7} x \left (a-b x^2\right )^{5/4}\right )}{11 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (60 a^2 d^2+232 a b c d+393 b^2 c^2\right )}{11 b}}{15 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right ) (10 a d+27 b c)}{15 b}}{19 b}-\frac {2 d x \left (a-b x^2\right )^{9/4} \left (c+d x^2\right )^2}{19 b}\)

Input:

Int[(a - b*x^2)^(5/4)*(c + d*x^2)^3,x]
 

Output:

(-2*d*x*(a - b*x^2)^(9/4)*(c + d*x^2)^2)/(19*b) + ((-2*d*(27*b*c + 10*a*d) 
*x*(a - b*x^2)^(9/4)*(c + d*x^2))/(15*b) + ((-2*d*(393*b^2*c^2 + 232*a*b*c 
*d + 60*a^2*d^2)*x*(a - b*x^2)^(9/4))/(11*b) + (3*(1045*b^3*c^3 + 570*a*b^ 
2*c^2*d + 228*a^2*b*c*d^2 + 40*a^3*d^3)*((2*x*(a - b*x^2)^(5/4))/7 + (5*a* 
((2*x*(a - b*x^2)^(1/4))/3 + (2*a^(3/2)*(1 - (b*x^2)/a)^(3/4)*EllipticF[Ar 
cSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*Sqrt[b]*(a - b*x^2)^(3/4))))/7))/(11*b 
))/(15*b))/(19*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 230

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2] 
))*EllipticF[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ 
[a, 0] && NegQ[b/a]
 

rule 231

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(3/4)/( 
a + b*x^2)^(3/4)   Int[1/(1 + b*(x^2/a))^(3/4), 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]

\[\int \left (-b \,x^{2}+a \right )^{\frac {5}{4}} \left (x^{2} d +c \right )^{3}d x\]

Input:

int((-b*x^2+a)^(5/4)*(d*x^2+c)^3,x)
 

Output:

int((-b*x^2+a)^(5/4)*(d*x^2+c)^3,x)
 

Fricas [F]

\[ \int \left (a-b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{3} \,d x } \] Input:

integrate((-b*x^2+a)^(5/4)*(d*x^2+c)^3,x, algorithm="fricas")
 

Output:

integral(-(b*d^3*x^8 + (3*b*c*d^2 - a*d^3)*x^6 + 3*(b*c^2*d - a*c*d^2)*x^4 
 - a*c^3 + (b*c^3 - 3*a*c^2*d)*x^2)*(-b*x^2 + a)^(1/4), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.04 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.96 \[ \int \left (a-b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=a^{\frac {5}{4}} c^{3} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + a^{\frac {5}{4}} c^{2} d x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + \frac {3 a^{\frac {5}{4}} c d^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} + \frac {a^{\frac {5}{4}} d^{3} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{7} - \frac {\sqrt [4]{a} b c^{3} x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{3} - \frac {3 \sqrt [4]{a} b c^{2} d x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} - \frac {3 \sqrt [4]{a} b c d^{2} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{7} - \frac {\sqrt [4]{a} b d^{3} x^{9} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {9}{2} \\ \frac {11}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{9} \] Input:

integrate((-b*x**2+a)**(5/4)*(d*x**2+c)**3,x)
 

Output:

a**(5/4)*c**3*x*hyper((-1/4, 1/2), (3/2,), b*x**2*exp_polar(2*I*pi)/a) + a 
**(5/4)*c**2*d*x**3*hyper((-1/4, 3/2), (5/2,), b*x**2*exp_polar(2*I*pi)/a) 
 + 3*a**(5/4)*c*d**2*x**5*hyper((-1/4, 5/2), (7/2,), b*x**2*exp_polar(2*I* 
pi)/a)/5 + a**(5/4)*d**3*x**7*hyper((-1/4, 7/2), (9/2,), b*x**2*exp_polar( 
2*I*pi)/a)/7 - a**(1/4)*b*c**3*x**3*hyper((-1/4, 3/2), (5/2,), b*x**2*exp_ 
polar(2*I*pi)/a)/3 - 3*a**(1/4)*b*c**2*d*x**5*hyper((-1/4, 5/2), (7/2,), b 
*x**2*exp_polar(2*I*pi)/a)/5 - 3*a**(1/4)*b*c*d**2*x**7*hyper((-1/4, 7/2), 
 (9/2,), b*x**2*exp_polar(2*I*pi)/a)/7 - a**(1/4)*b*d**3*x**9*hyper((-1/4, 
 9/2), (11/2,), b*x**2*exp_polar(2*I*pi)/a)/9
 

Maxima [F]

\[ \int \left (a-b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{3} \,d x } \] Input:

integrate((-b*x^2+a)^(5/4)*(d*x^2+c)^3,x, algorithm="maxima")
 

Output:

integrate((-b*x^2 + a)^(5/4)*(d*x^2 + c)^3, x)
 

Giac [F]

\[ \int \left (a-b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{3} \,d x } \] Input:

integrate((-b*x^2+a)^(5/4)*(d*x^2+c)^3,x, algorithm="giac")
 

Output:

integrate((-b*x^2 + a)^(5/4)*(d*x^2 + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a-b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=\int {\left (a-b\,x^2\right )}^{5/4}\,{\left (d\,x^2+c\right )}^3 \,d x \] Input:

int((a - b*x^2)^(5/4)*(c + d*x^2)^3,x)
 

Output:

int((a - b*x^2)^(5/4)*(c + d*x^2)^3, x)
 

Reduce [F]

\[ \int \left (a-b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=\frac {-200 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a^{4} d^{3} x -1140 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a^{3} b c \,d^{2} x -100 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a^{3} b \,d^{3} x^{3}-2850 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b^{2} c^{2} d x -570 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b^{2} c \,d^{2} x^{3}-70 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b^{2} d^{3} x^{5}+16720 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{3} c^{3} x +20520 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{3} c^{2} d \,x^{3}+12768 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{3} c \,d^{2} x^{5}+3080 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{3} d^{3} x^{7}-6270 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} b^{4} c^{3} x^{3}-11970 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} b^{4} c^{2} d \,x^{5}-8778 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} b^{4} c \,d^{2} x^{7}-2310 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} b^{4} d^{3} x^{9}+200 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}d x \right ) a^{5} d^{3}+1140 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}d x \right ) a^{4} b c \,d^{2}+2850 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}d x \right ) a^{3} b^{2} c^{2} d +5225 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}d x \right ) a^{2} b^{3} c^{3}}{21945 b^{3}} \] Input:

int((-b*x^2+a)^(5/4)*(d*x^2+c)^3,x)
 

Output:

( - 200*(a - b*x**2)**(1/4)*a**4*d**3*x - 1140*(a - b*x**2)**(1/4)*a**3*b* 
c*d**2*x - 100*(a - b*x**2)**(1/4)*a**3*b*d**3*x**3 - 2850*(a - b*x**2)**( 
1/4)*a**2*b**2*c**2*d*x - 570*(a - b*x**2)**(1/4)*a**2*b**2*c*d**2*x**3 - 
70*(a - b*x**2)**(1/4)*a**2*b**2*d**3*x**5 + 16720*(a - b*x**2)**(1/4)*a*b 
**3*c**3*x + 20520*(a - b*x**2)**(1/4)*a*b**3*c**2*d*x**3 + 12768*(a - b*x 
**2)**(1/4)*a*b**3*c*d**2*x**5 + 3080*(a - b*x**2)**(1/4)*a*b**3*d**3*x**7 
 - 6270*(a - b*x**2)**(1/4)*b**4*c**3*x**3 - 11970*(a - b*x**2)**(1/4)*b** 
4*c**2*d*x**5 - 8778*(a - b*x**2)**(1/4)*b**4*c*d**2*x**7 - 2310*(a - b*x* 
*2)**(1/4)*b**4*d**3*x**9 + 200*int((a - b*x**2)**(1/4)/(a - b*x**2),x)*a* 
*5*d**3 + 1140*int((a - b*x**2)**(1/4)/(a - b*x**2),x)*a**4*b*c*d**2 + 285 
0*int((a - b*x**2)**(1/4)/(a - b*x**2),x)*a**3*b**2*c**2*d + 5225*int((a - 
 b*x**2)**(1/4)/(a - b*x**2),x)*a**2*b**3*c**3)/(21945*b**3)