\(\int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx\) [1095]

Optimal result

Integrand size = 30, antiderivative size = 414 \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}-\frac {2 c^{3/4} (2 b c-5 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}+\frac {2 c^{3/4} (2 b c-5 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}} \] Output:

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

Mathematica [C] (verified)

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

Time = 11.13 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.35 \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=\frac {2 e (e x)^{3/2} \left (-7 a \left (c-d x^2\right )+7 a c \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+(2 b c-5 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{35 a b \sqrt {c-d x^2}} \] Input:

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

Output:

(2*e*(e*x)^(3/2)*(-7*a*(c - d*x^2) + 7*a*c*Sqrt[1 - (d*x^2)/c]*AppellF1[3/ 
4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + (2*b*c - 5*a*d)*x^2*Sqrt[1 - (d*x^ 
2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(35*a*b*Sqrt[c - 
 d*x^2])
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {368, 27, 978, 1054, 2009}

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.

\(\displaystyle \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle \frac {2 \int \frac {e^5 x^3 \sqrt {c-d x^2}}{a e^2-b e^2 x^2}d\sqrt {e x}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e \int \frac {e^3 x^3 \sqrt {c-d x^2}}{a e^2-b e^2 x^2}d\sqrt {e x}\)

\(\Big \downarrow \) 978

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

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 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 1054

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.51 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.28

method	result	size
risch	\(-\frac {2 x^{2} \sqrt {-x^{2} d +c}\, e^{3}}{5 b \sqrt {e x}}+\frac {\left (\frac {\left (5 a d -2 b c \right ) \sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \left (-\frac {2 \sqrt {c d}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {c d}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{b d \sqrt {-d e \,x^{3}+c e x}}+\frac {5 \left (a d -b c \right ) a \left (\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{b}\right ) e^{3} \sqrt {\left (-x^{2} d +c \right ) e x}}{5 b \sqrt {e x}\, \sqrt {-x^{2} d +c}}\)	\(528\)
elliptic	\(\text {Expression too large to display}\)	\(1003\)
default	\(\text {Expression too large to display}\)	\(1480\)

Input:

int((e*x)^(5/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-2/5*x^2*(-d*x^2+c)^(1/2)/b*e^3/(e*x)^(1/2)+1/5/b*((5*a*d-2*b*c)/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)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*(-2/d* 
(c*d)^(1/2)*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2 
))+1/d*(c*d)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2 
*2^(1/2)))+5*(a*d-b*c)*a/b*(1/2/b/d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)* 
(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(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/b/d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2)) 
^(1/2)*(-x*d/(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))))*e^3*((-d*x^2 
+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=\text {Timed out} \] Input:

integrate((e*x)^(5/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((e*x)**(5/2)*(-d*x**2+c)**(1/2)/(-b*x**2+a),x)
 

Output:

-Integral((e*x)**(5/2)*sqrt(c - d*x**2)/(-a + b*x**2), x)
 

Maxima [F]

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

integrate((e*x)^(5/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^(5/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(((e*x)^(5/2)*(c - d*x^2)^(1/2))/(a - b*x^2),x)
 

Output:

int(((e*x)^(5/2)*(c - d*x^2)^(1/2))/(a - b*x^2), x)
 

Reduce [F]

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

int((e*x)^(5/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x)
 

Output:

int((e*x)^(5/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x)