\(\int \frac {(e x)^{7/2}}{(a-b x^2) (c-d x^2)^{3/2}} \, dx\) [889]

Optimal result

Integrand size = 30, antiderivative size = 338 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-2 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b d^{5/4} (b c-a d) \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \]

[Out]

-c*e^3*(e*x)^(1/2)/d/(-a*d+b*c)/(-d*x^2+c)^(1/2)+c^(1/4)*(-2*a*d+b*c)*e^(7/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^
(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/b/d^(5/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)+a*c^(1/4)*e^(7/2)*EllipticPi(d^(1/4)*
(e*x)^(1/2)/c^(1/4)/e^(1/2),-b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)*(1-d*x^2/c)^(1/2)/b/d^(1/4)/(-a*d+b*c)/(-d*x^2
+c)^(1/2)+a*c^(1/4)*e^(7/2)*EllipticPi(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)*
(1-d*x^2/c)^(1/2)/b/d^(1/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {477, 481, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (b c-2 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b d^{5/4} \sqrt {c-d x^2} (b c-a d)}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}-\frac {c e^3 \sqrt {e x}}{d \sqrt {c-d x^2} (b c-a d)} \]

[In]

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

[Out]

-((c*e^3*Sqrt[e*x])/(d*(b*c - a*d)*Sqrt[c - d*x^2])) + (c^(1/4)*(b*c - 2*a*d)*e^(7/2)*Sqrt[1 - (d*x^2)/c]*Elli
pticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(b*d^(5/4)*(b*c - a*d)*Sqrt[c - d*x^2]) + (a*c^(1/4)
*e^(7/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^
(1/4)*Sqrt[e])], -1])/(b*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x^2]) + (a*c^(1/4)*e^(7/2)*Sqrt[1 - (d*x^2)/c]*Ellipti
cPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(b*d^(1/4)*(b*c -
a*d)*Sqrt[c - d*x^2])

Rule 227

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

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + b*(x^4/a)]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + b*(x^4/
a)], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-d/c, 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 481

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

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 1232

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[
a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 1233

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a)]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^8}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = -\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {e^3 \text {Subst}\left (\int \frac {a c-\frac {(b c-2 a d) x^4}{e^2}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{d (b c-a d)} \\ & = -\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\left (2 a^2 e^3\right ) \text {Subst}\left (\int \frac {1}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b (b c-a d)}+\frac {\left ((b c-2 a d) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b d (b c-a d)} \\ & = -\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b (b c-a d)}+\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b (b c-a d)}+\frac {\left ((b c-2 a d) e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b d (b c-a d) \sqrt {c-d x^2}} \\ & = -\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-2 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{5/4} (b c-a d) \sqrt {c-d x^2}}+\frac {\left (a e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b (b c-a d) \sqrt {c-d x^2}}+\frac {\left (a e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b (b c-a d) \sqrt {c-d x^2}} \\ & = -\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-2 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{5/4} (b c-a d) \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.44 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {e^3 \sqrt {e x} \left (-5 a c+5 a c \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+(-b c+2 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{5 a d (-b c+a d) \sqrt {c-d x^2}} \]

[In]

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

[Out]

-1/5*(e^3*Sqrt[e*x]*(-5*a*c + 5*a*c*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (-(
b*c) + 2*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(a*d*(-(b*c) + a*d)*S
qrt[c - d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(593\) vs. \(2(264)=528\).

Time = 3.09 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.76

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {e^{4} x c}{d \left (a d -b c \right ) \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) e^{4}}{d^{2} \sqrt {-d e \,x^{3}+c e x}\, b}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) c \,e^{4}}{2 d^{2} \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right )}+\frac {e^{4} a^{2} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) b \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {e^{4} a^{2} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) b \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-d \,x^{2}+c}}\)	\(594\)
default	\(-\frac {\left (2 F\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} d^{2} \sqrt {c d}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}-3 F\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b c d \sqrt {c d}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}+F\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, b^{2} c^{2} \sqrt {c d}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}+\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} b c \,d^{2}-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {c d}\, a^{2} d^{2}-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} b c \,d^{2}-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {c d}\, a^{2} d^{2}+2 a b c \,d^{2} x \sqrt {a b}-2 b^{2} c^{2} d x \sqrt {a b}\right ) e^{3} \sqrt {e x}}{2 d x \sqrt {-d \,x^{2}+c}\, \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {c d}\, b +\sqrt {a b}\, d \right ) \sqrt {a b}\, \left (a d -b c \right )}\)	\(815\)

[In]

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

[Out]

1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/d*e^4*x*c/(a*d-b*c)/(-(x^2-c/d)*d*e*x)^(1/2)+1/d^
2*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)
^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*e^4/b+1/2/d^2*(c*d)^(1/2)*(d*x/(c*d)^(
1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(
c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*c*e^4/(a*d-b*c)+1/2*e^4*a^2/(a*d-b*c)/b/(a*b)^(1/2)/d*(c*d)^(1/2
)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d
*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d
)^(1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2))-1/2*e^4*a^2/(a*d-b*c)/b/(a*b)^(1/2)/d*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1
/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(
1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2))
,1/2*2^(1/2)))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=- \int \frac {\left (e x\right )^{\frac {7}{2}}}{- a c \sqrt {c - d x^{2}} + a d x^{2} \sqrt {c - d x^{2}} + b c x^{2} \sqrt {c - d x^{2}} - b d x^{4} \sqrt {c - d x^{2}}}\, dx \]

[In]

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

[Out]

-Integral((e*x)**(7/2)/(-a*c*sqrt(c - d*x**2) + a*d*x**2*sqrt(c - d*x**2) + b*c*x**2*sqrt(c - d*x**2) - b*d*x*
*4*sqrt(c - d*x**2)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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