\(\int \frac {c+d x}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx\) [1376]

Optimal result

Integrand size = 25, antiderivative size = 218 \[ \int \frac {c+d x}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}-\frac {2 d \sqrt {a-b x^2}}{a e^2 \sqrt {e x}}-\frac {2 \sqrt [4]{b} d \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{\sqrt [4]{a} e^{5/2} \sqrt {a-b x^2}}+\frac {2 \sqrt [4]{b} \left (\sqrt {b} c+3 \sqrt {a} d\right ) \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{3 a^{3/4} e^{5/2} \sqrt {a-b x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.38 \[ \int \frac {c+d x}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=-\frac {2 x \sqrt {1-\frac {b x^2}{a}} \left (c \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},\frac {b x^2}{a}\right )+3 d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\frac {b x^2}{a}\right )\right )}{3 (e x)^{5/2} \sqrt {a-b x^2}} \] Input:

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

Output:

(-2*x*Sqrt[1 - (b*x^2)/a]*(c*Hypergeometric2F1[-3/4, 1/2, 1/4, (b*x^2)/a] 
+ 3*d*x*Hypergeometric2F1[-1/4, 1/2, 3/4, (b*x^2)/a]))/(3*(e*x)^(5/2)*Sqrt 
[a - b*x^2])
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {553, 27, 553, 27, 556, 555, 1513, 27, 765, 762, 1390, 1389, 327}

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.

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

\(\Big \downarrow \) 553

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 553

\(\displaystyle \frac {-\frac {2 \int -\frac {a b (c-3 d x)}{2 \sqrt {e x} \sqrt {a-b x^2}}dx}{a e}-\frac {6 d \sqrt {a-b x^2}}{e \sqrt {e x}}}{3 a e}-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \int \frac {c-3 d x}{\sqrt {e x} \sqrt {a-b x^2}}dx}{e}-\frac {6 d \sqrt {a-b x^2}}{e \sqrt {e x}}}{3 a e}-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 556

\(\displaystyle \frac {\frac {b \sqrt {x} \int \frac {c-3 d x}{\sqrt {x} \sqrt {a-b x^2}}dx}{e \sqrt {e x}}-\frac {6 d \sqrt {a-b x^2}}{e \sqrt {e x}}}{3 a e}-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 555

\(\displaystyle \frac {\frac {2 b \sqrt {x} \int \frac {c-3 d x}{\sqrt {a-b x^2}}d\sqrt {x}}{e \sqrt {e x}}-\frac {6 d \sqrt {a-b x^2}}{e \sqrt {e x}}}{3 a e}-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 1513

\(\displaystyle \frac {\frac {2 b \sqrt {x} \left (\left (\frac {3 \sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {x}-\frac {3 \sqrt {a} d \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^2}}d\sqrt {x}}{\sqrt {b}}\right )}{e \sqrt {e x}}-\frac {6 d \sqrt {a-b x^2}}{e \sqrt {e x}}}{3 a e}-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 b \sqrt {x} \left (\left (\frac {3 \sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {x}-\frac {3 d \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}}{\sqrt {b}}\right )}{e \sqrt {e x}}-\frac {6 d \sqrt {a-b x^2}}{e \sqrt {e x}}}{3 a e}-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\frac {2 b \sqrt {x} \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (\frac {3 \sqrt {a} d}{\sqrt {b}}+c\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {x}}{\sqrt {a-b x^2}}-\frac {3 d \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}}{\sqrt {b}}\right )}{e \sqrt {e x}}-\frac {6 d \sqrt {a-b x^2}}{e \sqrt {e x}}}{3 a e}-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\frac {2 b \sqrt {x} \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^2}{a}} \left (\frac {3 \sqrt {a} d}{\sqrt {b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 d \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}}{\sqrt {b}}\right )}{e \sqrt {e x}}-\frac {6 d \sqrt {a-b x^2}}{e \sqrt {e x}}}{3 a e}-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {\frac {2 b \sqrt {x} \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^2}{a}} \left (\frac {3 \sqrt {a} d}{\sqrt {b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 d \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {x}}{\sqrt {b} \sqrt {a-b x^2}}\right )}{e \sqrt {e x}}-\frac {6 d \sqrt {a-b x^2}}{e \sqrt {e x}}}{3 a e}-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {\frac {2 b \sqrt {x} \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^2}{a}} \left (\frac {3 \sqrt {a} d}{\sqrt {b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 \sqrt {a} d \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}d\sqrt {x}}{\sqrt {b} \sqrt {a-b x^2}}\right )}{e \sqrt {e x}}-\frac {6 d \sqrt {a-b x^2}}{e \sqrt {e x}}}{3 a e}-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {2 b \sqrt {x} \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^2}{a}} \left (\frac {3 \sqrt {a} d}{\sqrt {b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 a^{3/4} d \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^2}}\right )}{e \sqrt {e x}}-\frac {6 d \sqrt {a-b x^2}}{e \sqrt {e x}}}{3 a e}-\frac {2 c \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}\)

Input:

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

Output:

(-2*c*Sqrt[a - b*x^2])/(3*a*e*(e*x)^(3/2)) + ((-6*d*Sqrt[a - b*x^2])/(e*Sq 
rt[e*x]) + (2*b*Sqrt[x]*((-3*a^(3/4)*d*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSi 
n[(b^(1/4)*Sqrt[x])/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - b*x^2]) + (a^(1/4)*(c 
 + (3*Sqrt[a]*d)/Sqrt[b])*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[(b^(1/4)*Sq 
rt[x])/a^(1/4)], -1])/(b^(1/4)*Sqrt[a - b*x^2])))/(e*Sqrt[e*x]))/(3*a*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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 553

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp 
[1/(a*e*(m + 1))   Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 
2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[m, -1]
 

rule 555

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> 
Simp[2   Subst[Int[(f + g*x^2)/Sqrt[a + c*x^4], x], x, Sqrt[x]], x] /; Free 
Q[{a, c, f, g}, x]
 

rule 556

Int[((c_) + (d_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symb 
ol] :> Simp[Sqrt[x]/Sqrt[e*x]   Int[(c + d*x)/(Sqrt[x]*Sqrt[a + b*x^2]), x] 
, x] /; FreeQ[{a, b, c, d, e}, x]
 

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 1389

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

rule 1390

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

rule 1513

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

Maple [A] (verified)

Time = 1.31 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.31

method	result	size
default	\(-\frac {3 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a d x -\sqrt {2}\, \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) c x -6 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a d x -6 b d \,x^{3}-2 b c \,x^{2}+6 a d x +2 a c}{3 x \sqrt {-b \,x^{2}+a}\, a \,e^{2} \sqrt {e x}}\)	\(286\)
risch	\(-\frac {2 \sqrt {-b \,x^{2}+a}\, \left (3 d x +c \right )}{3 a x \,e^{2} \sqrt {e x}}+\frac {b \left (\frac {c \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}-\frac {3 d \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {-b e \,x^{3}+a e x}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) e x}}{3 a \,e^{2} \sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\)	\(325\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (-\frac {2 c \sqrt {-b e \,x^{3}+a e x}}{3 e^{3} a \,x^{2}}-\frac {2 \left (-b e \,x^{2}+a e \right ) d}{e^{3} a \sqrt {x \left (-b e \,x^{2}+a e \right )}}+\frac {c \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{3 a \,e^{2} \sqrt {-b e \,x^{3}+a e x}}-\frac {d \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{a \,e^{2} \sqrt {-b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\)	\(352\)

Input:

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

Output:

-1/3*(3*2^(1/2)*((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*((-b*x+(a*b)^(1/2))/ 
(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)*EllipticF(((b*x+(a*b)^(1/2))/( 
a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*d*x-2^(1/2)*(a*b)^(1/2)*((b*x+(a*b)^(1/2) 
)/(a*b)^(1/2))^(1/2)*((-b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1 
/2))^(1/2)*EllipticF(((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*c* 
x-6*2^(1/2)*((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*((-b*x+(a*b)^(1/2))/(a*b 
)^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)*EllipticE(((b*x+(a*b)^(1/2))/(a*b) 
^(1/2))^(1/2),1/2*2^(1/2))*a*d*x-6*b*d*x^3-2*b*c*x^2+6*a*d*x+2*a*c)/x/(-b* 
x^2+a)^(1/2)/a/e^2/(e*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.38 \[ \int \frac {c+d x}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left (\sqrt {-b e} c x^{2} {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right ) + 3 \, \sqrt {-b e} d x^{2} {\rm weierstrassZeta}\left (\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right )\right ) + \sqrt {-b x^{2} + a} {\left (3 \, d x + c\right )} \sqrt {e x}\right )}}{3 \, a e^{3} x^{2}} \] Input:

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

Output:

-2/3*(sqrt(-b*e)*c*x^2*weierstrassPInverse(4*a/b, 0, x) + 3*sqrt(-b*e)*d*x 
^2*weierstrassZeta(4*a/b, 0, weierstrassPInverse(4*a/b, 0, x)) + sqrt(-b*x 
^2 + a)*(3*d*x + c)*sqrt(e*x))/(a*e^3*x^2)
 

Sympy [A] (verification not implemented)

Time = 5.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.48 \[ \int \frac {c+d x}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=\frac {c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {d \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \] Input:

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

Output:

c*gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), b*x**2*exp_polar(2*I*pi)/a)/(2*sq 
rt(a)*e**(5/2)*x**(3/2)*gamma(1/4)) + d*gamma(-1/4)*hyper((-1/4, 1/2), (3/ 
4,), b*x**2*exp_polar(2*I*pi)/a)/(2*sqrt(a)*e**(5/2)*sqrt(x)*gamma(3/4))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {c+d x}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=\frac {\sqrt {e}\, \left (\left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{5}+a \,x^{3}}d x \right ) c +\left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{4}+a \,x^{2}}d x \right ) d \right )}{e^{3}} \] Input:

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

Output:

(sqrt(e)*(int((sqrt(x)*sqrt(a - b*x**2))/(a*x**3 - b*x**5),x)*c + int((sqr 
t(x)*sqrt(a - b*x**2))/(a*x**2 - b*x**4),x)*d))/e**3