\(\int (a+b x^2)^{5/4} (c+d x^2)^3 \, dx\) [415]

Optimal result

Integrand size = 21, antiderivative size = 303 \[ \int \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=\frac {2 a \left (1045 b^3 c^3-2 a d \left (285 b^2 c^2-114 a b c d+20 a^2 d^2\right )\right ) x \sqrt [4]{a+b x^2}}{4389 b^3}+\frac {2 \left (1045 c^3-\frac {2 a d \left (285 b^2 c^2-114 a b c d+20 a^2 d^2\right )}{b^3}\right ) x \left (a+b x^2\right )^{5/4}}{7315}+\frac {2 d \left (285 b^2 c^2-114 a b c d+20 a^2 d^2\right ) x \left (a+b x^2\right )^{9/4}}{1045 b^3}+\frac {2 d^2 (57 b c-10 a d) x^3 \left (a+b x^2\right )^{9/4}}{285 b^2}+\frac {2 d^3 x^5 \left (a+b x^2\right )^{9/4}}{19 b}+\frac {2 a^{5/2} \left (1045 b^3 c^3-2 a d \left (285 b^2 c^2-114 a b c d+20 a^2 d^2\right )\right ) \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{4389 b^{7/2} \left (a+b x^2\right )^{3/4}} \] 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^2-1 
14*a*b*c*d+285*b^2*c^2))*(1+b*x^2/a)^(3/4)*InverseJacobiAM(1/2*arctan(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 = 13.98 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.79 \[ \int \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=\frac {x \left (2 \left (a+b x^2\right ) \left (100 a^4 d^3-10 a^3 b d^2 \left (57 c+5 d x^2\right )+5 a^2 b^2 d \left (285 c^2+57 c d x^2+7 d^2 x^4\right )+3 b^4 x^2 \left (1045 c^3+1995 c^2 d x^2+1463 c d^2 x^4+385 d^3 x^6\right )+4 a b^3 \left (2090 c^3+2565 c^2 d x^2+1596 c d^2 x^4+385 d^3 x^6\right )\right )+5 a^2 \left (1045 b^3 c^3-570 a b^2 c^2 d+228 a^2 b c d^2-40 a^3 d^3\right ) \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{21945 b^3 \left (a+b x^2\right )^{3/4}} \] 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 + 2 
28*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 = 266, normalized size of antiderivative = 0.88, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {318, 27, 403, 27, 299, 211, 211, 231, 229}

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

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 231

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

\(\Big \downarrow \) 229

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

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[ArcTa 
n[(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 229

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) 
)*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[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 = 2.98 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.94 \[ \int \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=a^{\frac {5}{4}} c^{3} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} + a^{\frac {5}{4}} c^{2} d x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} + \frac {3 a^{\frac {5}{4}} c d^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5} + \frac {a^{\frac {5}{4}} d^{3} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7} + \frac {\sqrt [4]{a} b c^{3} x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} + \frac {3 \sqrt [4]{a} b c^{2} d x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5} + \frac {3 \sqrt [4]{a} b c d^{2} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7} + \frac {\sqrt [4]{a} b d^{3} x^{9} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {9}{2} \\ \frac {11}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{9} \] 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(I*pi)/a) + a** 
(5/4)*c**2*d*x**3*hyper((-1/4, 3/2), (5/2,), b*x**2*exp_polar(I*pi)/a) + 3 
*a**(5/4)*c*d**2*x**5*hyper((-1/4, 5/2), (7/2,), b*x**2*exp_polar(I*pi)/a) 
/5 + a**(5/4)*d**3*x**7*hyper((-1/4, 7/2), (9/2,), b*x**2*exp_polar(I*pi)/ 
a)/7 + a**(1/4)*b*c**3*x**3*hyper((-1/4, 3/2), (5/2,), b*x**2*exp_polar(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(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(I*pi)/a)/7 + a**(1/4)*b*d**3*x**9*hyper((-1/4, 9/2), (11/2,) 
, b*x**2*exp_polar(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. \[ \int \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^3 \, dx=\int {\left (b\,x^2+a\right )}^{5/4}\,{\left (d\,x^2+c\right )}^3 \,d x \] 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 - 2850*i 
nt((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)