\(\int \frac {(e x)^{5/4} (c+d x)}{\sqrt {c^2-d^2 x^2}} \, dx\) [585]

Optimal result

Integrand size = 29, antiderivative size = 204 \[ \int \frac {(e x)^{5/4} (c+d x)}{\sqrt {c^2-d^2 x^2}} \, dx=-\frac {4 c e \sqrt [4]{e x} \sqrt {c^2-d^2 x^2}}{5 d^2}-\frac {4 (e x)^{5/4} \sqrt {c^2-d^2 x^2}}{9 d}+\frac {4 c^3 e \sqrt [4]{e x} \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{2},\frac {9}{8},\frac {d^2 x^2}{c^2}\right )}{5 d^2 \sqrt {c^2-d^2 x^2}}+\frac {4 c^2 (e x)^{5/4} \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{8},\frac {13}{8},\frac {d^2 x^2}{c^2}\right )}{9 d \sqrt {c^2-d^2 x^2}} \] Output:

-4/5*c*e*(e*x)^(1/4)*(-d^2*x^2+c^2)^(1/2)/d^2-4/9*(e*x)^(5/4)*(-d^2*x^2+c^ 
2)^(1/2)/d+4/5*c^3*e*(e*x)^(1/4)*(1-d^2*x^2/c^2)^(1/2)*hypergeom([1/8, 1/2 
],[9/8],d^2*x^2/c^2)/d^2/(-d^2*x^2+c^2)^(1/2)+4/9*c^2*(e*x)^(5/4)*(1-d^2*x 
^2/c^2)^(1/2)*hypergeom([1/2, 5/8],[13/8],d^2*x^2/c^2)/d/(-d^2*x^2+c^2)^(1 
/2)
 

Mathematica [A] (verified)

Time = 10.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.72 \[ \int \frac {(e x)^{5/4} (c+d x)}{\sqrt {c^2-d^2 x^2}} \, dx=\frac {4 e \sqrt [4]{e x} \left (-9 c^3-5 c^2 d x+9 c d^2 x^2+5 d^3 x^3+9 c^3 \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{2},\frac {9}{8},\frac {d^2 x^2}{c^2}\right )+5 c^2 d x \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{8},\frac {13}{8},\frac {d^2 x^2}{c^2}\right )\right )}{45 d^2 \sqrt {c^2-d^2 x^2}} \] Input:

Integrate[((e*x)^(5/4)*(c + d*x))/Sqrt[c^2 - d^2*x^2],x]
 

Output:

(4*e*(e*x)^(1/4)*(-9*c^3 - 5*c^2*d*x + 9*c*d^2*x^2 + 5*d^3*x^3 + 9*c^3*Sqr 
t[1 - (d^2*x^2)/c^2]*Hypergeometric2F1[1/8, 1/2, 9/8, (d^2*x^2)/c^2] + 5*c 
^2*d*x*Sqrt[1 - (d^2*x^2)/c^2]*Hypergeometric2F1[1/2, 5/8, 13/8, (d^2*x^2) 
/c^2]))/(45*d^2*Sqrt[c^2 - d^2*x^2])
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(733\) vs. \(2(204)=408\).

Time = 1.40 (sec) , antiderivative size = 733, normalized size of antiderivative = 3.59, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {557, 262, 266, 767, 27, 889, 888, 2422}

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.

\(\displaystyle \int \frac {(e x)^{5/4} (c+d x)}{\sqrt {c^2-d^2 x^2}} \, dx\)

\(\Big \downarrow \) 557

\(\displaystyle c \int \frac {(e x)^{5/4}}{\sqrt {c^2-d^2 x^2}}dx+\frac {d \int \frac {(e x)^{9/4}}{\sqrt {c^2-d^2 x^2}}dx}{e}\)

\(\Big \downarrow \) 262

\(\displaystyle c \left (\frac {c^2 e^2 \int \frac {1}{(e x)^{3/4} \sqrt {c^2-d^2 x^2}}dx}{5 d^2}-\frac {4 e \sqrt [4]{e x} \sqrt {c^2-d^2 x^2}}{5 d^2}\right )+\frac {d \left (\frac {5 c^2 e^2 \int \frac {\sqrt [4]{e x}}{\sqrt {c^2-d^2 x^2}}dx}{9 d^2}-\frac {4 e (e x)^{5/4} \sqrt {c^2-d^2 x^2}}{9 d^2}\right )}{e}\)

\(\Big \downarrow \) 266

\(\displaystyle c \left (\frac {4 c^2 e \int \frac {1}{\sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}}{5 d^2}-\frac {4 e \sqrt [4]{e x} \sqrt {c^2-d^2 x^2}}{5 d^2}\right )+\frac {d \left (\frac {20 c^2 e \int \frac {e x}{\sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}}{9 d^2}-\frac {4 e (e x)^{5/4} \sqrt {c^2-d^2 x^2}}{9 d^2}\right )}{e}\)

\(\Big \downarrow \) 767

\(\displaystyle \frac {d \left (\frac {20 c^2 e \int \frac {e x}{\sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}}{9 d^2}-\frac {4 e (e x)^{5/4} \sqrt {c^2-d^2 x^2}}{9 d^2}\right )}{e}+c \left (\frac {4 c^2 e \left (\frac {1}{2} \int \frac {\sqrt {e}-\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}}{\sqrt {e} \sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}+\frac {1}{2} \int \frac {\sqrt {e}+\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}}{\sqrt {e} \sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}\right )}{5 d^2}-\frac {4 e \sqrt [4]{e x} \sqrt {c^2-d^2 x^2}}{5 d^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {20 c^2 e \int \frac {e x}{\sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}}{9 d^2}-\frac {4 e (e x)^{5/4} \sqrt {c^2-d^2 x^2}}{9 d^2}\right )}{e}+c \left (\frac {4 c^2 e \left (\frac {\int \frac {\sqrt {e}-\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}}{\sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}}{2 \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}}{\sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}}{2 \sqrt {e}}\right )}{5 d^2}-\frac {4 e \sqrt [4]{e x} \sqrt {c^2-d^2 x^2}}{5 d^2}\right )\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {d \left (\frac {20 c^2 e \sqrt {1-\frac {d^2 x^2}{c^2}} \int \frac {e x}{\sqrt {1-\frac {d^2 x^2}{c^2}}}d\sqrt [4]{e x}}{9 d^2 \sqrt {c^2-d^2 x^2}}-\frac {4 e (e x)^{5/4} \sqrt {c^2-d^2 x^2}}{9 d^2}\right )}{e}+c \left (\frac {4 c^2 e \left (\frac {\int \frac {\sqrt {e}-\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}}{\sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}}{2 \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}}{\sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}}{2 \sqrt {e}}\right )}{5 d^2}-\frac {4 e \sqrt [4]{e x} \sqrt {c^2-d^2 x^2}}{5 d^2}\right )\)

\(\Big \downarrow \) 888

\(\displaystyle c \left (\frac {4 c^2 e \left (\frac {\int \frac {\sqrt {e}-\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}}{\sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}}{2 \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}}{\sqrt {c^2-d^2 x^2}}d\sqrt [4]{e x}}{2 \sqrt {e}}\right )}{5 d^2}-\frac {4 e \sqrt [4]{e x} \sqrt {c^2-d^2 x^2}}{5 d^2}\right )+\frac {d \left (\frac {4 c^2 e (e x)^{5/4} \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{8},\frac {13}{8},\frac {d^2 x^2}{c^2}\right )}{9 d^2 \sqrt {c^2-d^2 x^2}}-\frac {4 e (e x)^{5/4} \sqrt {c^2-d^2 x^2}}{9 d^2}\right )}{e}\)

\(\Big \downarrow \) 2422

\(\displaystyle c \left (\frac {4 c^2 e \left (\frac {\sqrt {d} (e x)^{3/4} \sqrt {\frac {c^2 e^2-d^2 e^2 x^2}{\sqrt {-c^2} d e^2 x}} \sqrt {\frac {\sqrt [4]{-c^2} \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}+\sqrt {e}\right )^2}{\sqrt {d} \sqrt {e} \sqrt {e x}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt [4]{-c^2} \left (\frac {\sqrt {2} d x e}{\sqrt {-c^2}}+\sqrt {2} e-\frac {2 \sqrt {d} \sqrt {e x} \sqrt {e}}{\sqrt [4]{-c^2}}\right )}{\sqrt {d} \sqrt {e} \sqrt {e x}}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [4]{-c^2} \sqrt {c^2-d^2 x^2} \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}+\sqrt {e}\right )}-\frac {\sqrt {d} (e x)^{3/4} \sqrt {\frac {c^2 e^2-d^2 e^2 x^2}{\sqrt {-c^2} d e^2 x}} \sqrt {-\frac {\sqrt [4]{-c^2} \left (\sqrt {e}-\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}\right )^2}{\sqrt {d} \sqrt {e} \sqrt {e x}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt [4]{-c^2} \left (\frac {\sqrt {2} d x e}{\sqrt {-c^2}}+\sqrt {2} e+\frac {2 \sqrt {d} \sqrt {e x} \sqrt {e}}{\sqrt [4]{-c^2}}\right )}{\sqrt {d} \sqrt {e} \sqrt {e x}}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [4]{-c^2} \sqrt {c^2-d^2 x^2} \left (\sqrt {e}-\frac {\sqrt {d} \sqrt {e x}}{\sqrt [4]{-c^2}}\right )}\right )}{5 d^2}-\frac {4 e \sqrt [4]{e x} \sqrt {c^2-d^2 x^2}}{5 d^2}\right )+\frac {d \left (\frac {4 c^2 e (e x)^{5/4} \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{8},\frac {13}{8},\frac {d^2 x^2}{c^2}\right )}{9 d^2 \sqrt {c^2-d^2 x^2}}-\frac {4 e (e x)^{5/4} \sqrt {c^2-d^2 x^2}}{9 d^2}\right )}{e}\)

Input:

Int[((e*x)^(5/4)*(c + d*x))/Sqrt[c^2 - d^2*x^2],x]
 

Output:

c*((-4*e*(e*x)^(1/4)*Sqrt[c^2 - d^2*x^2])/(5*d^2) + (4*c^2*e*((Sqrt[d]*(e* 
x)^(3/4)*Sqrt[(c^2*e^2 - d^2*e^2*x^2)/(Sqrt[-c^2]*d*e^2*x)]*Sqrt[((-c^2)^( 
1/4)*(Sqrt[e] + (Sqrt[d]*Sqrt[e*x])/(-c^2)^(1/4))^2)/(Sqrt[d]*Sqrt[e]*Sqrt 
[e*x])]*EllipticF[ArcSin[Sqrt[-(((-c^2)^(1/4)*(Sqrt[2]*e + (Sqrt[2]*d*e*x) 
/Sqrt[-c^2] - (2*Sqrt[d]*Sqrt[e]*Sqrt[e*x])/(-c^2)^(1/4)))/(Sqrt[d]*Sqrt[e 
]*Sqrt[e*x]))]/2], -2*(1 - Sqrt[2])])/(2*Sqrt[2 + Sqrt[2]]*(-c^2)^(1/4)*Sq 
rt[c^2 - d^2*x^2]*(Sqrt[e] + (Sqrt[d]*Sqrt[e*x])/(-c^2)^(1/4))) - (Sqrt[d] 
*(e*x)^(3/4)*Sqrt[(c^2*e^2 - d^2*e^2*x^2)/(Sqrt[-c^2]*d*e^2*x)]*Sqrt[-(((- 
c^2)^(1/4)*(Sqrt[e] - (Sqrt[d]*Sqrt[e*x])/(-c^2)^(1/4))^2)/(Sqrt[d]*Sqrt[e 
]*Sqrt[e*x]))]*EllipticF[ArcSin[Sqrt[((-c^2)^(1/4)*(Sqrt[2]*e + (Sqrt[2]*d 
*e*x)/Sqrt[-c^2] + (2*Sqrt[d]*Sqrt[e]*Sqrt[e*x])/(-c^2)^(1/4)))/(Sqrt[d]*S 
qrt[e]*Sqrt[e*x])]/2], -2*(1 - Sqrt[2])])/(2*Sqrt[2 + Sqrt[2]]*(-c^2)^(1/4 
)*Sqrt[c^2 - d^2*x^2]*(Sqrt[e] - (Sqrt[d]*Sqrt[e*x])/(-c^2)^(1/4)))))/(5*d 
^2)) + (d*((-4*e*(e*x)^(5/4)*Sqrt[c^2 - d^2*x^2])/(9*d^2) + (4*c^2*e*(e*x) 
^(5/4)*Sqrt[1 - (d^2*x^2)/c^2]*Hypergeometric2F1[1/2, 5/8, 13/8, (d^2*x^2) 
/c^2])/(9*d^2*Sqrt[c^2 - d^2*x^2])))/e
 

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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 557

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c   Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e   Int[(e*x)^( 
m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
 

rule 767

Int[1/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> Simp[1/2   Int[(1 - Rt[b/a, 4 
]*x^2)/Sqrt[a + b*x^8], x], x] + Simp[1/2   Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a 
 + b*x^8], x], x] /; FreeQ[{a, b}, x]
 

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

rule 2422

Int[((c_) + (d_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> Simp[(-c) 
*d*x^3*Sqrt[-(c - d*x^2)^2/(c*d*x^2)]*(Sqrt[(-d^2)*((a + b*x^8)/(b*c^2*x^4) 
)]/(Sqrt[2 + Sqrt[2]]*(c - d*x^2)*Sqrt[a + b*x^8]))*EllipticF[ArcSin[(1/2)* 
Sqrt[(Sqrt[2]*c^2 + 2*c*d*x^2 + Sqrt[2]*d^2*x^4)/(c*d*x^2)]], -2*(1 - Sqrt[ 
2])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^4 - a*d^4, 0]
 

Maple [F]

\[\int \frac {\left (e x \right )^{\frac {5}{4}} \left (d x +c \right )}{\sqrt {-d^{2} x^{2}+c^{2}}}d x\]

Input:

int((e*x)^(5/4)*(d*x+c)/(-d^2*x^2+c^2)^(1/2),x)
 

Output:

int((e*x)^(5/4)*(d*x+c)/(-d^2*x^2+c^2)^(1/2),x)
 

Fricas [F]

\[ \int \frac {(e x)^{5/4} (c+d x)}{\sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{\frac {5}{4}}}{\sqrt {-d^{2} x^{2} + c^{2}}} \,d x } \] Input:

integrate((e*x)^(5/4)*(d*x+c)/(-d^2*x^2+c^2)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-d^2*x^2 + c^2)*(e*x)^(1/4)*e*x/(d*x - c), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 19.83 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.46 \[ \int \frac {(e x)^{5/4} (c+d x)}{\sqrt {c^2-d^2 x^2}} \, dx=\frac {e^{\frac {5}{4}} x^{\frac {9}{4}} \Gamma \left (\frac {9}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{8} \\ \frac {17}{8} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )}}{2 \Gamma \left (\frac {17}{8}\right )} + \frac {d e^{\frac {5}{4}} x^{\frac {13}{4}} \Gamma \left (\frac {13}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {13}{8} \\ \frac {21}{8} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )}}{2 c \Gamma \left (\frac {21}{8}\right )} \] Input:

integrate((e*x)**(5/4)*(d*x+c)/(-d**2*x**2+c**2)**(1/2),x)
 

Output:

e**(5/4)*x**(9/4)*gamma(9/8)*hyper((1/2, 9/8), (17/8,), d**2*x**2*exp_pola 
r(2*I*pi)/c**2)/(2*gamma(17/8)) + d*e**(5/4)*x**(13/4)*gamma(13/8)*hyper(( 
1/2, 13/8), (21/8,), d**2*x**2*exp_polar(2*I*pi)/c**2)/(2*c*gamma(21/8))
 

Maxima [F]

\[ \int \frac {(e x)^{5/4} (c+d x)}{\sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{\frac {5}{4}}}{\sqrt {-d^{2} x^{2} + c^{2}}} \,d x } \] Input:

integrate((e*x)^(5/4)*(d*x+c)/(-d^2*x^2+c^2)^(1/2),x, algorithm="maxima")
 

Output:

integrate((d*x + c)*(e*x)^(5/4)/sqrt(-d^2*x^2 + c^2), x)
 

Giac [F]

\[ \int \frac {(e x)^{5/4} (c+d x)}{\sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{\frac {5}{4}}}{\sqrt {-d^{2} x^{2} + c^{2}}} \,d x } \] Input:

integrate((e*x)^(5/4)*(d*x+c)/(-d^2*x^2+c^2)^(1/2),x, algorithm="giac")
 

Output:

integrate((d*x + c)*(e*x)^(5/4)/sqrt(-d^2*x^2 + c^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/4} (c+d x)}{\sqrt {c^2-d^2 x^2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/4}\,\left (c+d\,x\right )}{\sqrt {c^2-d^2\,x^2}} \,d x \] Input:

int(((e*x)^(5/4)*(c + d*x))/(c^2 - d^2*x^2)^(1/2),x)
 

Output:

int(((e*x)^(5/4)*(c + d*x))/(c^2 - d^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(e x)^{5/4} (c+d x)}{\sqrt {c^2-d^2 x^2}} \, dx=\frac {e^{\frac {5}{4}} \left (-36 x^{\frac {1}{4}} \sqrt {-d^{2} x^{2}+c^{2}}\, c -20 x^{\frac {5}{4}} \sqrt {-d^{2} x^{2}+c^{2}}\, d +9 \left (\int \frac {\sqrt {-d^{2} x^{2}+c^{2}}}{x^{\frac {3}{4}} c^{2}-x^{\frac {11}{4}} d^{2}}d x \right ) c^{3}+25 \left (\int \frac {\sqrt {-d^{2} x^{2}+c^{2}}\, x}{x^{\frac {3}{4}} c^{2}-x^{\frac {11}{4}} d^{2}}d x \right ) c^{2} d \right )}{45 d^{2}} \] Input:

int((e*x)^(5/4)*(d*x+c)/(-d^2*x^2+c^2)^(1/2),x)
 

Output:

(e**(1/4)*e*( - 36*x**(1/4)*sqrt(c**2 - d**2*x**2)*c - 20*x**(1/4)*sqrt(c* 
*2 - d**2*x**2)*d*x + 9*int(sqrt(c**2 - d**2*x**2)/(x**(3/4)*c**2 - x**(3/ 
4)*d**2*x**2),x)*c**3 + 25*int((sqrt(c**2 - d**2*x**2)*x)/(x**(3/4)*c**2 - 
 x**(3/4)*d**2*x**2),x)*c**2*d))/(45*d**2)