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

3.9.65.1 Optimal result

Integrand size = 30, antiderivative size = 372 \[ \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=\frac {2 (2 b c-7 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{21 b^2 d}-\frac {2 e (e x)^{5/2} \sqrt {c-d x^2}}{7 b}-\frac {2 \sqrt [4]{c} \left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{21 b^3 d^{5/4} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} (b c-a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} (b c-a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}} \]

output

-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)*Ellipti 
cPi(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)
 

3.9.65.2 Mathematica [C] (verified)

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

Time = 11.18 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.50 \[ \int \frac {(e x)^{7/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=\frac {2 e^3 \sqrt {e x} \left (-5 a \left (c-d x^2\right ) \left (-2 b c+7 a d+3 b d x^2\right )+5 a c (-2 b c+7 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+\left (2 b^2 c^2+14 a b c d-21 a^2 d^2\right ) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{105 a b^2 d \sqrt {c-d x^2}} \]

input

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

output

(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]*AppellF1[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])
 

3.9.65.3 Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {368, 27, 978, 1052, 1021, 765, 762, 925, 27, 1543, 1542}

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.

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

\(\Big \downarrow \) 368

\(\displaystyle \frac {2 \int \frac {e^6 x^4 \sqrt {c-d x^2}}{a e^2-b e^2 x^2}d\sqrt {e x}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e \int \frac {e^4 x^4 \sqrt {c-d x^2}}{a e^2-b e^2 x^2}d\sqrt {e x}\)

\(\Big \downarrow \) 978

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

\(\Big \downarrow \) 1052

\(\displaystyle 2 e \left (\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (2 b c-7 a d)}{3 b d}-\frac {e^2 \int \frac {a c (2 b c-7 a d) e^2-\left (2 b^2 c^2+14 a b d c-21 a^2 d^2\right ) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b d}}{7 b}-\frac {(e x)^{5/2} \sqrt {c-d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 1021

\(\displaystyle 2 e \left (\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (2 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{b}-\frac {21 a^2 d e^2 (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}\right )}{3 b d}}{7 b}-\frac {(e x)^{5/2} \sqrt {c-d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 765

\(\displaystyle 2 e \left (\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (2 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{b \sqrt {c-d x^2}}-\frac {21 a^2 d e^2 (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}\right )}{3 b d}}{7 b}-\frac {(e x)^{5/2} \sqrt {c-d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 762

\(\displaystyle 2 e \left (\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (2 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {21 a^2 d e^2 (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}\right )}{3 b d}}{7 b}-\frac {(e x)^{5/2} \sqrt {c-d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 925

\(\displaystyle 2 e \left (\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (2 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {21 a^2 d e^2 (b c-a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )}{b}\right )}{3 b d}}{7 b}-\frac {(e x)^{5/2} \sqrt {c-d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e \left (\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (2 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {21 a^2 d e^2 (b c-a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )}{b}\right )}{3 b d}}{7 b}-\frac {(e x)^{5/2} \sqrt {c-d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 1543

\(\displaystyle 2 e \left (\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (2 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {21 a^2 d e^2 (b c-a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )}{b}\right )}{3 b d}}{7 b}-\frac {(e x)^{5/2} \sqrt {c-d x^2}}{7 b}\right )\)

\(\Big \downarrow \) 1542

\(\displaystyle 2 e \left (\frac {\frac {e^2 \sqrt {e x} \sqrt {c-d x^2} (2 b c-7 a d)}{3 b d}-\frac {e^2 \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {21 a^2 d e^2 (b c-a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{b}\right )}{3 b d}}{7 b}-\frac {(e x)^{5/2} \sqrt {c-d x^2}}{7 b}\right )\)

input

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

output

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

3.9.65.3.1 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 368

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[x^(k*(m + 1) 
 - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], 
 x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m 
] && IntegerQ[p]
 

rule 762

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

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[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 925

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 
1/(2*c)   Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[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 978

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] - Simp[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 1021

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x 
_)^(n_)]), x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^n], x], x] + Simp[(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 1052

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] - Simp[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 1542

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 1543

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 
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]
 

3.9.65.4 Maple [A] (verified)

Time = 4.10 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.41

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

input

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

output

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

3.9.65.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.9.65.6 Sympy [F]

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

input

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

output

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

3.9.65.7 Maxima [F]

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

input

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

output

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

3.9.65.8 Giac [F]

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

input

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

output

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

3.9.65.9 Mupad [F(-1)]

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

input

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

output

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