3.9.65 \(\int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{a-b x^2} \, dx\) [865]

Optimal. Leaf size=372 \[ \frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}-\frac {2 \sqrt [4]{c} \left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) e^{7/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{21 b^3 d^{5/4} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} (b c-a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} (b c-a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}} \]

[Out]

-2/7*e*(e*x)^(5/2)*(-d*x^2+c)^(1/2)/b+2/21*(-7*a*d+2*b*c)*e^3*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b^2/d-2/21*c^(1/4)*
(-21*a^2*d^2+14*a*b*c*d+2*b^2*c^2)*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^3/d^(5/4)/(-d*x^2+c)^(1/2)+a*c^(1/4)*(-a*d+b*c)*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^3/d^(1/4)/(-d*x^2+c)^(1/2)+a*c^(1/4)*(-a*d+b*c)*e^(7/2)*Ell
ipticPi(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^3/d^(1/4)/(
-d*x^2+c)^(1/2)

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Rubi [A]
time = 0.54, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {477, 489, 596, 537, 230, 227, 418, 1233, 1232} \begin {gather*} -\frac {2 \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{21 b^3 d^{5/4} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {2 e^3 \sqrt {e x} \sqrt {c-d x^2} (2 b c-7 a d)}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2*b*c - 7*a*d)*e^3*Sqrt[e*x]*Sqrt[c - d*x^2])/(21*b^2*d) - (2*e*(e*x)^(5/2)*Sqrt[c - d*x^2])/(7*b) - (2*c^
(1/4)*(2*b^2*c^2 + 14*a*b*c*d - 21*a^2*d^2)*e^(7/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(
c^(1/4)*Sqrt[e])], -1])/(21*b^3*d^(5/4)*Sqrt[c - d*x^2]) + (a*c^(1/4)*(b*c - a*d)*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^3*d^
(1/4)*Sqrt[c - d*x^2]) + (a*c^(1/4)*(b*c - a*d)*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^3*d^(1/4)*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 489

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Dist[e^n/(b*(m + n*(p +
q) + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b
*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] &&
GtQ[m - n + 1, 0] && 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 596

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

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*} \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{a-b x^2} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^8 \sqrt {c-\frac {d x^4}{e^2}}}{a-\frac {b x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}+\frac {(2 e) \text {Subst}\left (\int \frac {x^4 \left (5 a c+\frac {(2 b c-7 a d) x^4}{e^2}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{7 b}\\ &=\frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}-\frac {\left (2 e^5\right ) \text {Subst}\left (\int \frac {\frac {a c (2 b c-7 a d)}{e^2}-\frac {\left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) x^4}{e^4}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b^2 d}\\ &=\frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}+\frac {\left (2 a^2 (b c-a d) 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^3}-\frac {\left (2 \left (2 b^2 c^2+14 a b c d-21 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 )}{21 b^3 d}\\ &=\frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}+\frac {\left (a (b c-a d) 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^3}+\frac {\left (a (b c-a d) 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^3}-\frac {\left (2 \left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) 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 )}{21 b^3 d \sqrt {c-d x^2}}\\ &=\frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}-\frac {2 \sqrt [4]{c} \left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) 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 )}{21 b^3 d^{5/4} \sqrt {c-d x^2}}+\frac {\left (a (b c-a d) 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^3 \sqrt {c-d x^2}}+\frac {\left (a (b c-a d) 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^3 \sqrt {c-d x^2}}\\ &=\frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}-\frac {2 \sqrt [4]{c} \left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) 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 )}{21 b^3 d^{5/4} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} (b c-a d) 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^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} (b c-a d) 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^3 \sqrt [4]{d} \sqrt {c-d x^2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 10.18, size = 187, normalized size = 0.50 \begin {gather*} \frac {2 e^3 \sqrt {e x} \left (-5 a \left (c-d x^2\right ) \left (-2 b c+7 a d+3 b d x^2\right )+5 a c (-2 b c+7 a d) \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+\left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) x^2 \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{105 a b^2 d \sqrt {c-d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e^3*Sqrt[e*x]*(-5*a*(c - d*x^2)*(-2*b*c + 7*a*d + 3*b*d*x^2) + 5*a*c*(-2*b*c + 7*a*d)*Sqrt[1 - (d*x^2)/c]*A
ppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (2*b^2*c^2 + 14*a*b*c*d - 21*a^2*d^2)*x^2*Sqrt[1 - (d*x^2)/c
]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(105*a*b^2*d*Sqrt[c - d*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1467\) vs. \(2(288)=576\).
time = 0.22, size = 1468, normalized size = 3.95

method	result	size
risch	\(-\frac {2 \left (3 b d \,x^{2}+7 a d -2 b c \right ) \sqrt {-d \,x^{2}+c}\, x \,e^{4}}{21 d \,b^{2} \sqrt {e x}}+\frac {\left (\frac {\left (21 a^{2} d^{2}-14 a b c d -2 b^{2} c^{2}\right ) \sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{b d \sqrt {-d e \,x^{3}+c e x}}+\frac {21 a^{2} \left (a d -b c \right ) d \left (\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{b}\right ) e^{4} \sqrt {\left (-d \,x^{2}+c \right ) e x}}{21 d \,b^{2} \sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\)	\(523\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (-\frac {2 e^{3} x^{2} \sqrt {-d e \,x^{3}+c e x}}{7 b}-\frac {2 \left (\frac {\left (a d -b c \right ) e^{4}}{b^{2}}+\frac {5 e^{4} c}{7 b}\right ) \sqrt {-d e \,x^{3}+c e x}}{3 d e}+\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}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} e^{4}}{\sqrt {-d e \,x^{3}+c e x}\, b^{3}}-\frac {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}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a \,e^{4} c}{3 d \sqrt {-d e \,x^{3}+c e x}\, b^{2}}-\frac {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}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) c^{2} e^{4}}{21 d^{2} \sqrt {-d e \,x^{3}+c e x}\, b}+\frac {a^{3} e^{4} \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}}}\, \EllipticPi \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 b^{3} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {a^{2} e^{4} \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}}}\, \EllipticPi \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 ) c}{2 b^{2} \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 {a^{3} e^{4} \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}}}\, \EllipticPi \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 b^{3} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {a^{2} e^{4} \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}}}\, \EllipticPi \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 ) c}{2 b^{2} \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}}\)	\(1005\)
default	\(\text {Expression too large to display}\)	\(1468\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(7/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-1/42*e^3*(e*x)^(1/2)/b^2/d*(42*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^3*d^3*(
a*b)^(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)*(-d*x/(c*d)^(1/2))^(1/
2)*(c*d)^(1/2)-70*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^2*b*c*d^2*(a*b)^(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)*(-d*x/(c*d)^(1/2))^(1/2)*(c*d)^(
1/2)+24*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a*b^2*c^2*d*(a*b)^(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)*(-d*x/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)+4*Ell
ipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*b^3*c^3*(a*b)^(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)*(-d*x/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)+21*EllipticPi(((d*x+
(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b+(a*b)^(1/2)*d),1/2*2^(1/2))*2^(1/2)*a^3*b*c*d^3*(
(d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)-21*Ellipt
icPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b+(a*b)^(1/2)*d),1/2*2^(1/2))*2^(1/2)*a^
3*d^3*(a*b)^(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)*(-d*x/(c*d)^(1/
2))^(1/2)*(c*d)^(1/2)-21*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b+(a*b)^(
1/2)*d),1/2*2^(1/2))*2^(1/2)*a^2*b^2*c^2*d^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)*(-d*x/(c*d)^(1/2))^(1/2)+21*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^
(1/2)*b+(a*b)^(1/2)*d),1/2*2^(1/2))*2^(1/2)*a^2*b*c*d^2*(a*b)^(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)*(-d*x/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)-21*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)
^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*2^(1/2)*a^3*b*c*d^3*((d*x+(c*d)^(1/2))/
(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)-21*EllipticPi(((d*x+(c*d)^(
1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*2^(1/2)*a^3*d^3*(a*b)^(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)*(-d*x/(c*d)^(1/2))^(1/2)*(c*d)^(1
/2)+21*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2
))*2^(1/2)*a^2*b^2*c^2*d^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)*(-d*x/
(c*d)^(1/2))^(1/2)+21*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2
)*d),1/2*2^(1/2))*2^(1/2)*a^2*b*c*d^2*(a*b)^(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)*(-d*x/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)+12*a*b^2*d^4*x^5*(a*b)^(1/2)-12*b^3*c*d^3*x^5*(a*b)^(1/2
)+28*a^2*b*d^4*x^3*(a*b)^(1/2)-48*a*b^2*c*d^3*x^3*(a*b)^(1/2)+20*b^3*c^2*d^2*x^3*(a*b)^(1/2)-28*a^2*b*c*d^3*x*
(a*b)^(1/2)+36*a*b^2*c^2*d^2*x*(a*b)^(1/2)-8*b^3*c^3*d*x*(a*b)^(1/2))/x/(-d*x^2+c)^(1/2)/(a*b)^(1/2)/((c*d)^(1
/2)*b+(a*b)^(1/2)*d)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(7/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(7/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\left (e x\right )^{\frac {7}{2}} \sqrt {c - d x^{2}}}{- a + b x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(7/2)*(-d*x**2+c)**(1/2)/(-b*x**2+a),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(7/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x\right )}^{7/2}\,\sqrt {c-d\,x^2}}{a-b\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e*x)^(7/2)*(c - d*x^2)^(1/2))/(a - b*x^2),x)

[Out]

int(((e*x)^(7/2)*(c - d*x^2)^(1/2))/(a - b*x^2), x)

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