\(\int \frac {(e x)^{7/2} (a+b x^2)^2}{(c+d x^2)^{3/2}} \, dx\) [849]

Optimal result

Integrand size = 28, antiderivative size = 296 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {5 \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{231 d^4}-\frac {\left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e (e x)^{5/2} \sqrt {c+d x^2}}{77 c d^3}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d^2 e}-\frac {5 c^{3/4} \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{462 d^{17/4} \sqrt {c+d x^2}} \]

[Out]

(-a*d+b*c)^2*(e*x)^(9/2)/c/d^2/e/(d*x^2+c)^(1/2)-1/77*(77*a^2*d^2-198*a*b*c*d+117*b^2*c^2)*e*(e*x)^(5/2)*(d*x^
2+c)^(1/2)/c/d^3+2/11*b^2*(e*x)^(9/2)*(d*x^2+c)^(1/2)/d^2/e+5/231*(77*a^2*d^2-198*a*b*c*d+117*b^2*c^2)*e^3*(e*
x)^(1/2)*(d*x^2+c)^(1/2)/d^4-5/462*c^(3/4)*(77*a^2*d^2-198*a*b*c*d+117*b^2*c^2)*e^(7/2)*(cos(2*arctan(d^(1/4)*
(e*x)^(1/2)/c^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))*EllipticF(sin(2*arct
an(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2))*(c^(1/2)+x*d^(1/2))*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1
/2)/d^(17/4)/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {474, 470, 327, 335, 226} \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {5 c^{3/4} e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{462 d^{17/4} \sqrt {c+d x^2}}+\frac {5 e^3 \sqrt {e x} \sqrt {c+d x^2} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right )}{231 d^4}-\frac {e (e x)^{5/2} \sqrt {c+d x^2} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right )}{77 c d^3}+\frac {(e x)^{9/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d^2 e} \]

[In]

Int[((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]

[Out]

((b*c - a*d)^2*(e*x)^(9/2))/(c*d^2*e*Sqrt[c + d*x^2]) + (5*(117*b^2*c^2 - 198*a*b*c*d + 77*a^2*d^2)*e^3*Sqrt[e
*x]*Sqrt[c + d*x^2])/(231*d^4) - ((117*b^2*c^2 - 198*a*b*c*d + 77*a^2*d^2)*e*(e*x)^(5/2)*Sqrt[c + d*x^2])/(77*
c*d^3) + (2*b^2*(e*x)^(9/2)*Sqrt[c + d*x^2])/(11*d^2*e) - (5*c^(3/4)*(117*b^2*c^2 - 198*a*b*c*d + 77*a^2*d^2)*
e^(7/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])
/(c^(1/4)*Sqrt[e])], 1/2])/(462*d^(17/4)*Sqrt[c + d*x^2])

Rule 226

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]

Rule 327

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

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 470

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

Rule 474

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

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\int \frac {(e x)^{7/2} \left (\frac {1}{2} \left (-2 a^2 d^2+9 (b c-a d)^2\right )-b^2 c d x^2\right )}{\sqrt {c+d x^2}} \, dx}{c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d^2 e}-\frac {\left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) \int \frac {(e x)^{7/2}}{\sqrt {c+d x^2}} \, dx}{22 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e (e x)^{5/2} \sqrt {c+d x^2}}{77 c d^3}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d^2 e}+\frac {\left (5 \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^2\right ) \int \frac {(e x)^{3/2}}{\sqrt {c+d x^2}} \, dx}{154 d^3} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {5 \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{231 d^4}-\frac {\left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e (e x)^{5/2} \sqrt {c+d x^2}}{77 c d^3}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d^2 e}-\frac {\left (5 c \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{462 d^4} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {5 \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{231 d^4}-\frac {\left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e (e x)^{5/2} \sqrt {c+d x^2}}{77 c d^3}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d^2 e}-\frac {\left (5 c \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{231 d^4} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {5 \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{231 d^4}-\frac {\left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e (e x)^{5/2} \sqrt {c+d x^2}}{77 c d^3}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d^2 e}-\frac {5 c^{3/4} \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{462 d^{17/4} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.19 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.76 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {e^3 \sqrt {e x} \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} \left (77 a^2 d^2 \left (5 c+2 d x^2\right )+66 a b d \left (-15 c^2-6 c d x^2+2 d^2 x^4\right )+3 b^2 \left (195 c^3+78 c^2 d x^2-26 c d^2 x^4+14 d^3 x^6\right )\right )-5 i c \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \sqrt {x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )\right )}{231 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^4 \sqrt {c+d x^2}} \]

[In]

Integrate[((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]

[Out]

(e^3*Sqrt[e*x]*(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(77*a^2*d^2*(5*c + 2*d*x^2) + 66*a*b*d*(-15*c^2 - 6*c*d*x^2 + 2*d^2*
x^4) + 3*b^2*(195*c^3 + 78*c^2*d*x^2 - 26*c*d^2*x^4 + 14*d^3*x^6)) - (5*I)*c*(117*b^2*c^2 - 198*a*b*c*d + 77*a
^2*d^2)*Sqrt[1 + c/(d*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/(231*Sqrt[(I
*Sqrt[c])/Sqrt[d]]*d^4*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.69 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.38

method	result	size
default	\(-\frac {e^{3} \sqrt {e x}\, \left (-84 b^{2} d^{4} x^{7}+385 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-990 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +585 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-264 a b \,d^{4} x^{5}+156 b^{2} c \,d^{3} x^{5}-308 a^{2} d^{4} x^{3}+792 x^{3} d^{3} b a c -468 b^{2} c^{2} d^{2} x^{3}-770 a^{2} c \,d^{3} x +1980 a b \,c^{2} d^{2} x -1170 b^{2} d x \,c^{3}\right )}{462 x \sqrt {d \,x^{2}+c}\, d^{5}}\)	\(407\)
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {e^{4} x c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} e^{3} x^{4} \sqrt {d e \,x^{3}+c e x}}{11 d^{2}}+\frac {2 \left (\frac {\left (2 a d -b c \right ) b \,e^{4}}{d^{2}}-\frac {9 b^{2} e^{4} c}{11 d^{2}}\right ) x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d e}+\frac {2 \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) e^{4}}{d^{3}}-\frac {5 \left (\frac {\left (2 a d -b c \right ) b \,e^{4}}{d^{2}}-\frac {9 b^{2} e^{4} c}{11 d^{2}}\right ) c}{7 d}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (-\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) e^{4}}{2 d^{4}}-\frac {\left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) e^{4}}{d^{3}}-\frac {5 \left (\frac {\left (2 a d -b c \right ) b \,e^{4}}{d^{2}}-\frac {9 b^{2} e^{4} c}{11 d^{2}}\right ) c}{7 d}\right ) c}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\)	\(458\)
risch	\(\frac {2 \left (21 b^{2} d^{2} x^{4}+66 x^{2} a b \,d^{2}-60 x^{2} b^{2} c d +77 a^{2} d^{2}-264 a b c d +177 b^{2} c^{2}\right ) x \sqrt {d \,x^{2}+c}\, e^{4}}{231 d^{4} \sqrt {e x}}-\frac {c \left (\frac {308 a^{2} d \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}+\frac {408 b^{2} c^{2} \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}-\frac {726 a b c \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}-231 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {x}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e^{4} \sqrt {e x \left (d \,x^{2}+c \right )}}{231 d^{4} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(610\)

[In]

int((e*x)^(7/2)*(b*x^2+a)^2/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/462*e^3/x*(e*x)^(1/2)*(-84*b^2*d^4*x^7+385*(-c*d)^(1/2)*2^(1/2)*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*((-
d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1
/2),1/2*2^(1/2))*a^2*c*d^2-990*(-c*d)^(1/2)*2^(1/2)*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*((-d*x+(-c*d)^(1/2
))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2)
)*a*b*c^2*d+585*(-c*d)^(1/2)*2^(1/2)*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2)
)^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^3-264*a
*b*d^4*x^5+156*b^2*c*d^3*x^5-308*a^2*d^4*x^3+792*x^3*d^3*b*a*c-468*b^2*c^2*d^2*x^3-770*a^2*c*d^3*x+1980*a*b*c^
2*d^2*x-1170*b^2*d*x*c^3)/(d*x^2+c)^(1/2)/d^5

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.77 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {5 \, {\left ({\left (117 \, b^{2} c^{3} d - 198 \, a b c^{2} d^{2} + 77 \, a^{2} c d^{3}\right )} e^{3} x^{2} + {\left (117 \, b^{2} c^{4} - 198 \, a b c^{3} d + 77 \, a^{2} c^{2} d^{2}\right )} e^{3}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (42 \, b^{2} d^{4} e^{3} x^{6} - 6 \, {\left (13 \, b^{2} c d^{3} - 22 \, a b d^{4}\right )} e^{3} x^{4} + 2 \, {\left (117 \, b^{2} c^{2} d^{2} - 198 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} e^{3} x^{2} + 5 \, {\left (117 \, b^{2} c^{3} d - 198 \, a b c^{2} d^{2} + 77 \, a^{2} c d^{3}\right )} e^{3}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{231 \, {\left (d^{6} x^{2} + c d^{5}\right )}} \]

[In]

integrate((e*x)^(7/2)*(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

-1/231*(5*((117*b^2*c^3*d - 198*a*b*c^2*d^2 + 77*a^2*c*d^3)*e^3*x^2 + (117*b^2*c^4 - 198*a*b*c^3*d + 77*a^2*c^
2*d^2)*e^3)*sqrt(d*e)*weierstrassPInverse(-4*c/d, 0, x) - (42*b^2*d^4*e^3*x^6 - 6*(13*b^2*c*d^3 - 22*a*b*d^4)*
e^3*x^4 + 2*(117*b^2*c^2*d^2 - 198*a*b*c*d^3 + 77*a^2*d^4)*e^3*x^2 + 5*(117*b^2*c^3*d - 198*a*b*c^2*d^2 + 77*a
^2*c*d^3)*e^3)*sqrt(d*x^2 + c)*sqrt(e*x))/(d^6*x^2 + c*d^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x)**(7/2)*(b*x**2+a)**2/(d*x**2+c)**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x)^(7/2)*(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(7/2)/(d*x^2 + c)^(3/2), x)

Giac [F]

\[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x)^(7/2)*(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(7/2)/(d*x^2 + c)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]

[In]

int(((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x)

[Out]

int(((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2), x)