\(\int \frac {(a+b x^2)^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx\) [1044]

Optimal result

Integrand size = 26, antiderivative size = 210 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\frac {2}{21} \left (7 b^2+\frac {a d (14 b c-a d)}{c^2}\right ) \sqrt {x} \sqrt {c+d x^2}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}-\frac {2 a (14 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^{3/2}}+\frac {2 \left (7 b^2 c^2+a d (14 b c-a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{21 c^{5/4} \sqrt [4]{d} \sqrt {c+d x^2}} \] Output:

2/21*(7*b^2+a*d*(-a*d+14*b*c)/c^2)*x^(1/2)*(d*x^2+c)^(1/2)-2/7*a^2*(d*x^2+ 
c)^(3/2)/c/x^(7/2)-2/21*a*(-a*d+14*b*c)*(d*x^2+c)^(3/2)/c^2/x^(3/2)+2/21*( 
7*b^2*c^2+a*d*(-a*d+14*b*c))*(c^(1/2)+d^(1/2)*x)*((d*x^2+c)/(c^(1/2)+d^(1/ 
2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(d^(1/4)*x^(1/2)/c^(1/4)),1/2*2^(1/ 
2))/c^(5/4)/d^(1/4)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.15 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\frac {2 \left (\left (c+d x^2\right ) \left (-14 a b c x^2+7 b^2 c x^4-a^2 \left (3 c+2 d x^2\right )\right )+\frac {2 i \left (7 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x^{9/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{21 c x^{7/2} \sqrt {c+d x^2}} \] Input:

Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]
 

Output:

(2*((c + d*x^2)*(-14*a*b*c*x^2 + 7*b^2*c*x^4 - a^2*(3*c + 2*d*x^2)) + ((2* 
I)*(7*b^2*c^2 + 14*a*b*c*d - a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^(9/2)*Elliptic 
F[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[c])/Sqrt 
[d]]))/(21*c*x^(7/2)*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {365, 27, 359, 248, 266, 761}

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx\)

\(\Big \downarrow \) 365

\(\displaystyle \frac {2 \int \frac {\left (7 b^2 c x^2+a (14 b c-a d)\right ) \sqrt {d x^2+c}}{2 x^{5/2}}dx}{7 c}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (7 b^2 c x^2+a (14 b c-a d)\right ) \sqrt {d x^2+c}}{x^{5/2}}dx}{7 c}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 359

\(\displaystyle \frac {\frac {\left (a d (14 b c-a d)+7 b^2 c^2\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {x}}dx}{c}-\frac {2 a \left (c+d x^2\right )^{3/2} (14 b c-a d)}{3 c x^{3/2}}}{7 c}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {\frac {\left (a d (14 b c-a d)+7 b^2 c^2\right ) \left (\frac {2}{3} c \int \frac {1}{\sqrt {x} \sqrt {d x^2+c}}dx+\frac {2}{3} \sqrt {x} \sqrt {c+d x^2}\right )}{c}-\frac {2 a \left (c+d x^2\right )^{3/2} (14 b c-a d)}{3 c x^{3/2}}}{7 c}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\frac {\left (a d (14 b c-a d)+7 b^2 c^2\right ) \left (\frac {4}{3} c \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {x}+\frac {2}{3} \sqrt {x} \sqrt {c+d x^2}\right )}{c}-\frac {2 a \left (c+d x^2\right )^{3/2} (14 b c-a d)}{3 c x^{3/2}}}{7 c}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {\left (a d (14 b c-a d)+7 b^2 c^2\right ) \left (\frac {2 c^{3/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{d} \sqrt {c+d x^2}}+\frac {2}{3} \sqrt {x} \sqrt {c+d x^2}\right )}{c}-\frac {2 a \left (c+d x^2\right )^{3/2} (14 b c-a d)}{3 c x^{3/2}}}{7 c}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}\)

Input:

Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]
 

Output:

(-2*a^2*(c + d*x^2)^(3/2))/(7*c*x^(7/2)) + ((-2*a*(14*b*c - a*d)*(c + d*x^ 
2)^(3/2))/(3*c*x^(3/2)) + ((7*b^2*c^2 + a*d*(14*b*c - a*d))*((2*Sqrt[x]*Sq 
rt[c + d*x^2])/3 + (2*c^(3/4)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt 
[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[x])/c^(1/4)], 1/2])/( 
3*d^(1/4)*Sqrt[c + d*x^2])))/c)/(7*c)
 

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 248

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) 
  Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ 
p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 359

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + 
Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] 
&& LtQ[m, -1] &&  !ILtQ[p, -1]
 

rule 365

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x 
_Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] 
- Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p 
+ 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.99

method	result	size
risch	\(-\frac {2 \sqrt {x^{2} d +c}\, \left (-7 b^{2} c \,x^{4}+2 a^{2} d \,x^{2}+14 a b c \,x^{2}+3 a^{2} c \right )}{21 x^{\frac {7}{2}} c}-\frac {2 \left (a^{2} d^{2}-14 a b c d -7 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x \left (x^{2} d +c \right )}}{21 c d \sqrt {d \,x^{3}+c x}\, \sqrt {x}\, \sqrt {x^{2} d +c}}\)	\(208\)
elliptic	\(\frac {\sqrt {x \left (x^{2} d +c \right )}\, \left (-\frac {2 a^{2} \sqrt {d \,x^{3}+c x}}{7 x^{4}}-\frac {4 a \left (a d +7 b c \right ) \sqrt {d \,x^{3}+c x}}{21 c \,x^{2}}+\frac {2 b^{2} \sqrt {d \,x^{3}+c x}}{3}+\frac {\left (2 a b d +\frac {2 b^{2} c}{3}-\frac {2 d a \left (a d +7 b c \right )}{21 c}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d \,x^{3}+c x}}\right )}{\sqrt {x}\, \sqrt {x^{2} d +c}}\)	\(223\)
default	\(-\frac {2 \left (\sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2} x^{3}-14 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d \,x^{3}-7 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2} x^{3}-7 b^{2} c \,d^{2} x^{6}+2 a^{2} d^{3} x^{4}+14 a b c \,d^{2} x^{4}-7 b^{2} c^{2} d \,x^{4}+5 a^{2} c \,d^{2} x^{2}+14 a b \,c^{2} d \,x^{2}+3 a^{2} c^{2} d \right )}{21 \sqrt {x^{2} d +c}\, x^{\frac {7}{2}} c d}\)	\(385\)

Input:

int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(9/2),x,method=_RETURNVERBOSE)
 

Output:

-2/21*(d*x^2+c)^(1/2)*(-7*b^2*c*x^4+2*a^2*d*x^2+14*a*b*c*x^2+3*a^2*c)/x^(7 
/2)/c-2/21*(a^2*d^2-14*a*b*c*d-7*b^2*c^2)/c*(-c*d)^(1/2)/d*((x+(-c*d)^(1/2 
)/d)/(-c*d)^(1/2)*d)^(1/2)*(-2*(x-(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*(- 
d/(-c*d)^(1/2)*x)^(1/2)/(d*x^3+c*x)^(1/2)*EllipticF(((x+(-c*d)^(1/2)/d)/(- 
c*d)^(1/2)*d)^(1/2),1/2*2^(1/2))*(x*(d*x^2+c))^(1/2)/x^(1/2)/(d*x^2+c)^(1/ 
2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.49 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\frac {2 \, {\left (2 \, {\left (7 \, b^{2} c^{2} + 14 \, a b c d - a^{2} d^{2}\right )} \sqrt {d} x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (7 \, b^{2} c d x^{4} - 3 \, a^{2} c d - 2 \, {\left (7 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )}}{21 \, c d x^{4}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(9/2),x, algorithm="fricas")
 

Output:

2/21*(2*(7*b^2*c^2 + 14*a*b*c*d - a^2*d^2)*sqrt(d)*x^4*weierstrassPInverse 
(-4*c/d, 0, x) + (7*b^2*c*d*x^4 - 3*a^2*c*d - 2*(7*a*b*c*d + a^2*d^2)*x^2) 
*sqrt(d*x^2 + c)*sqrt(x))/(c*d*x^4)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 10.16 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\frac {a^{2} \sqrt {c} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} + \frac {a b \sqrt {c} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {b^{2} \sqrt {c} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} \] Input:

integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**(9/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{x^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(9/2),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{x^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(9/2),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{x^{9/2}} \,d x \] Input:

int(((a + b*x^2)^2*(c + d*x^2)^(1/2))/x^(9/2),x)
 

Output:

int(((a + b*x^2)^2*(c + d*x^2)^(1/2))/x^(9/2), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\frac {-\frac {2 \sqrt {d \,x^{2}+c}\, a^{2} d^{2}}{5}+\frac {8 \sqrt {d \,x^{2}+c}\, a b c d}{5}-4 \sqrt {d \,x^{2}+c}\, a b \,d^{2} x^{2}+\frac {4 \sqrt {d \,x^{2}+c}\, b^{2} c^{2}}{5}-\frac {4 \sqrt {d \,x^{2}+c}\, b^{2} c d \,x^{2}}{3}+\frac {2 \sqrt {d \,x^{2}+c}\, b^{2} d^{2} x^{4}}{3}-\frac {2 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{7}+c \,x^{5}}d x \right ) a^{2} c \,d^{2} x^{3}}{5}+\frac {28 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{7}+c \,x^{5}}d x \right ) a b \,c^{2} d \,x^{3}}{5}+\frac {14 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{7}+c \,x^{5}}d x \right ) b^{2} c^{3} x^{3}}{5}}{\sqrt {x}\, d^{2} x^{3}} \] Input:

int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(9/2),x)
 

Output:

(2*( - 3*sqrt(c + d*x**2)*a**2*d**2 + 12*sqrt(c + d*x**2)*a*b*c*d - 30*sqr 
t(c + d*x**2)*a*b*d**2*x**2 + 6*sqrt(c + d*x**2)*b**2*c**2 - 10*sqrt(c + d 
*x**2)*b**2*c*d*x**2 + 5*sqrt(c + d*x**2)*b**2*d**2*x**4 - 3*sqrt(x)*int(( 
sqrt(x)*sqrt(c + d*x**2))/(c*x**5 + d*x**7),x)*a**2*c*d**2*x**3 + 42*sqrt( 
x)*int((sqrt(x)*sqrt(c + d*x**2))/(c*x**5 + d*x**7),x)*a*b*c**2*d*x**3 + 2 
1*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c*x**5 + d*x**7),x)*b**2*c**3*x* 
*3))/(15*sqrt(x)*d**2*x**3)