\(\int \frac {(e x)^{7/2} (a+b x^2)^2}{(c+d x^2)^{5/2}} \, dx\) [857]

Optimal result

Integrand size = 28, antiderivative size = 302 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}} \]

[Out]

1/3*(-a*d+b*c)^2*(e*x)^(9/2)/c/d^2/e/(d*x^2+c)^(3/2)+1/14*(7*a^2*d^2-42*a*b*c*d+39*b^2*c^2)*e*(e*x)^(5/2)/c/d^
3/(d*x^2+c)^(1/2)+2/7*b^2*(e*x)^(9/2)/d^2/e/(d*x^2+c)^(1/2)-5/42*(7*a^2*d^2-42*a*b*c*d+39*b^2*c^2)*e^3*(e*x)^(
1/2)*(d*x^2+c)^(1/2)/c/d^4+5/84*(7*a^2*d^2-42*a*b*c*d+39*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*arctan(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)/c^(1/4)/d^
(17/4)/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {474, 470, 294, 327, 335, 226} \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {5 e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}}-\frac {5 e^3 \sqrt {e x} \sqrt {c+d x^2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{42 c d^4}+\frac {e (e x)^{5/2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{14 c d^3 \sqrt {c+d x^2}}+\frac {(e x)^{9/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}} \]

[In]

Int[((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]

[Out]

((b*c - a*d)^2*(e*x)^(9/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) + ((39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^2)*e*(e*x)^(5/
2))/(14*c*d^3*Sqrt[c + d*x^2]) + (2*b^2*(e*x)^(9/2))/(7*d^2*e*Sqrt[c + d*x^2]) - (5*(39*b^2*c^2 - 42*a*b*c*d +
7*a^2*d^2)*e^3*Sqrt[e*x]*Sqrt[c + d*x^2])/(42*c*d^4) + (5*(39*b^2*c^2 - 42*a*b*c*d + 7*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)*Sqr
t[e])], 1/2])/(84*c^(1/4)*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 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

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}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {(e x)^{7/2} \left (-\frac {3}{2} \left (2 a^2 d^2-3 (b c-a d)^2\right )-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) \int \frac {(e x)^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{14 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^2\right ) \int \frac {(e x)^{3/2}}{\sqrt {c+d x^2}} \, dx}{28 c d^3} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{84 d^4} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 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 )}{42 d^4} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {5 \left (39 b^2 c^2-42 a b c d+7 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 )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.21 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.74 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(e x)^{7/2} \left (\frac {\sqrt {x} \left (-7 a^2 d^2 \left (5 c+7 d x^2\right )+14 a b d \left (15 c^2+21 c d x^2+4 d^2 x^4\right )-b^2 \left (195 c^3+273 c^2 d x^2+52 c d^2 x^4-12 d^3 x^6\right )\right )}{d^4 \left (c+d x^2\right )}+\frac {5 i \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^4}\right )}{42 x^{7/2} \sqrt {c+d x^2}} \]

[In]

Integrate[((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]

[Out]

((e*x)^(7/2)*((Sqrt[x]*(-7*a^2*d^2*(5*c + 7*d*x^2) + 14*a*b*d*(15*c^2 + 21*c*d*x^2 + 4*d^2*x^4) - b^2*(195*c^3
+ 273*c^2*d*x^2 + 52*c*d^2*x^4 - 12*d^3*x^6)))/(d^4*(c + d*x^2)) + ((5*I)*(39*b^2*c^2 - 42*a*b*c*d + 7*a^2*d^
2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d
]]*d^4)))/(42*x^(7/2)*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 4.54 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.39

method	result	size
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {c \,e^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d^{6} \left (x^{2}+\frac {c}{d}\right )^{2}}-\frac {e^{4} x \left (7 a^{2} d^{2}-26 a b c d +19 b^{2} c^{2}\right )}{6 d^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} e^{3} x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d^{3}}+\frac {2 \left (\frac {2 b \left (a d -b c \right ) e^{4}}{d^{3}}-\frac {5 b^{2} e^{4} c}{7 d^{3}}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (\frac {\left (a^{2} d^{2}-4 a b c d +3 b^{2} c^{2}\right ) e^{4}}{d^{4}}-\frac {e^{4} \left (7 a^{2} d^{2}-26 a b c d +19 b^{2} c^{2}\right )}{12 d^{4}}-\frac {\left (\frac {2 b \left (a d -b c \right ) e^{4}}{d^{3}}-\frac {5 b^{2} e^{4} c}{7 d^{3}}\right ) c}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\)	\(420\)
default	\(\frac {\left (35 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a^{2} d^{3} x^{2}-210 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a b c \,d^{2} x^{2}+195 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, b^{2} c^{2} d \,x^{2}+24 b^{2} d^{4} x^{7}+35 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-210 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +195 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}+112 a b \,d^{4} x^{5}-104 b^{2} c \,d^{3} x^{5}-98 a^{2} d^{4} x^{3}+588 x^{3} d^{3} b a c -546 b^{2} c^{2} d^{2} x^{3}-70 a^{2} c \,d^{3} x +420 a b \,c^{2} d^{2} x -390 b^{2} d x \,c^{3}\right ) e^{3} \sqrt {e x}}{84 x \,d^{5} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}\)	\(696\)
risch	\(\frac {2 b \left (3 b d \,x^{2}+14 a d -19 b c \right ) x \sqrt {d \,x^{2}+c}\, e^{4}}{21 d^{4} \sqrt {e x}}+\frac {\left (\frac {21 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 {82 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 {98 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}}-42 c \left (a^{2} d^{2}-3 a b c d +2 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 )+21 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\sqrt {d e \,x^{3}+c e x}}{3 c e \,d^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {5 x}{6 c^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {5 \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 )}{12 c^{2} d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e^{4} \sqrt {e x \left (d \,x^{2}+c \right )}}{21 d^{4} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(769\)

[In]

int((e*x)^(7/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

(e*x*(d*x^2+c))^(1/2)/e/x*(e*x)^(1/2)/(d*x^2+c)^(1/2)*(1/3*c*e^3/d^6*(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*e*x^3+c*e*
x)^(1/2)/(x^2+c/d)^2-1/6/d^4*e^4*x*(7*a^2*d^2-26*a*b*c*d+19*b^2*c^2)/((x^2+c/d)*d*e*x)^(1/2)+2/7*b^2/d^3*e^3*x
^2*(d*e*x^3+c*e*x)^(1/2)+2/3*(2*b/d^3*(a*d-b*c)*e^4-5/7*b^2/d^3*e^4*c)/d/e*(d*e*x^3+c*e*x)^(1/2)+((a^2*d^2-4*a
*b*c*d+3*b^2*c^2)*e^4/d^4-1/12/d^4*e^4*(7*a^2*d^2-26*a*b*c*d+19*b^2*c^2)-1/3*(2*b/d^3*(a*d-b*c)*e^4-5/7*b^2/d^
3*e^4*c)/d*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)*(-x/(-c*d)^(1/2)*d)^(1/2)/(d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2*
2^(1/2)))

Fricas [C] (verification not implemented)

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

Time = 0.12 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.91 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {5 \, {\left ({\left (39 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} e^{3} x^{4} + 2 \, {\left (39 \, b^{2} c^{3} d - 42 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} e^{3} x^{2} + {\left (39 \, b^{2} c^{4} - 42 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} e^{3}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (12 \, b^{2} d^{4} e^{3} x^{6} - 4 \, {\left (13 \, b^{2} c d^{3} - 14 \, a b d^{4}\right )} e^{3} x^{4} - 7 \, {\left (39 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} e^{3} x^{2} - 5 \, {\left (39 \, b^{2} c^{3} d - 42 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} e^{3}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{42 \, {\left (d^{7} x^{4} + 2 \, c d^{6} x^{2} + c^{2} d^{5}\right )}} \]

[In]

integrate((e*x)^(7/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

1/42*(5*((39*b^2*c^2*d^2 - 42*a*b*c*d^3 + 7*a^2*d^4)*e^3*x^4 + 2*(39*b^2*c^3*d - 42*a*b*c^2*d^2 + 7*a^2*c*d^3)
*e^3*x^2 + (39*b^2*c^4 - 42*a*b*c^3*d + 7*a^2*c^2*d^2)*e^3)*sqrt(d*e)*weierstrassPInverse(-4*c/d, 0, x) + (12*
b^2*d^4*e^3*x^6 - 4*(13*b^2*c*d^3 - 14*a*b*d^4)*e^3*x^4 - 7*(39*b^2*c^2*d^2 - 42*a*b*c*d^3 + 7*a^2*d^4)*e^3*x^
2 - 5*(39*b^2*c^3*d - 42*a*b*c^2*d^2 + 7*a^2*c*d^3)*e^3)*sqrt(d*x^2 + c)*sqrt(e*x))/(d^7*x^4 + 2*c*d^6*x^2 + c
^2*d^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x)**(7/2)*(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x)^(7/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(7/2)/(d*x^2 + c)^(5/2), x)

Giac [F]

\[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x)^(7/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(7/2)/(d*x^2 + c)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int(((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x)

[Out]

int(((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x)