\(\int \frac {(c+d x)^2}{\sqrt {e x} \sqrt {a-b x^2}} \, dx\) [1381]

Optimal result

Integrand size = 27, antiderivative size = 203 \[ \int \frac {(c+d x)^2}{\sqrt {e x} \sqrt {a-b x^2}} \, dx=-\frac {2 d^2 \sqrt {e x} \sqrt {a-b x^2}}{3 b e}+\frac {4 a^{3/4} c d \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} \sqrt {e} \sqrt {a-b x^2}}+\frac {2 \sqrt [4]{a} \left (3 b c^2-6 \sqrt {a} \sqrt {b} c d+a d^2\right ) \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{3 b^{5/4} \sqrt {e} \sqrt {a-b x^2}} \] Output:

-2/3*d^2*(e*x)^(1/2)*(-b*x^2+a)^(1/2)/b/e+4*a^(3/4)*c*d*(1-b*x^2/a)^(1/2)* 
EllipticE(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I)/b^(3/4)/e^(1/2)/(-b*x^2+a 
)^(1/2)+2/3*a^(1/4)*(3*b*c^2-6*a^(1/2)*b^(1/2)*c*d+a*d^2)*(1-b*x^2/a)^(1/2 
)*EllipticF(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I)/b^(5/4)/e^(1/2)/(-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 = 10.10 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.62 \[ \int \frac {(c+d x)^2}{\sqrt {e x} \sqrt {a-b x^2}} \, dx=\frac {2 x \left (\left (3 b c^2+a d^2\right ) \sqrt {1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^2}{a}\right )+d \left (-a d+b d x^2+2 b c x \sqrt {1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^2}{a}\right )\right )\right )}{3 b \sqrt {e x} \sqrt {a-b x^2}} \] Input:

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

Output:

(2*x*((3*b*c^2 + a*d^2)*Sqrt[1 - (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/ 
4, (b*x^2)/a] + d*(-(a*d) + b*d*x^2 + 2*b*c*x*Sqrt[1 - (b*x^2)/a]*Hypergeo 
metric2F1[1/2, 3/4, 7/4, (b*x^2)/a])))/(3*b*Sqrt[e*x]*Sqrt[a - b*x^2])
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.97, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {559, 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.

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

\(\Big \downarrow \) 559

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 556

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

\(\Big \downarrow \) 555

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

\(\Big \downarrow \) 1513

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(-2*d^2*Sqrt[e*x]*Sqrt[a - b*x^2])/(3*b*e) + (2*Sqrt[x]*((6*a^(3/4)*b^(1/4 
)*c*d*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[(b^(1/4)*Sqrt[x])/a^(1/4)], -1] 
)/Sqrt[a - b*x^2] + (a^(1/4)*(3*b*c^2 - 6*Sqrt[a]*Sqrt[b]*c*d + a*d^2)*Sqr 
t[1 - (b*x^2)/a]*EllipticF[ArcSin[(b^(1/4)*Sqrt[x])/a^(1/4)], -1])/(b^(1/4 
)*Sqrt[a - b*x^2])))/(3*b*Sqrt[e*x])
 

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 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 559

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( 
m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1))   Int[(e*x)^m*(a + b* 
x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 
)*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && IGtQ[n, 1] &&  !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(321\) vs. \(2(155)=310\).

Time = 0.95 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.59

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (-\frac {2 d^{2} \sqrt {-b e \,x^{3}+a e x}}{3 b e}+\frac {\left (c^{2}+\frac {d^{2} a}{3 b}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}+\frac {2 c d \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {-b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\)	\(322\)
default	\(\frac {\sqrt {2}\, \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}+3 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2} \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}+6 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a b c d -12 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a b c d +2 x^{3} b^{2} d^{2}-2 a b \,d^{2} x}{3 \sqrt {-b \,x^{2}+a}\, \sqrt {e x}\, b^{2}}\)	\(362\)
risch	\(-\frac {2 d^{2} x \sqrt {-b \,x^{2}+a}}{3 b \sqrt {e x}}+\frac {\left (\frac {a \,d^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}+\frac {3 c^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-b e \,x^{3}+a e x}}+\frac {6 d c \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {-b e \,x^{3}+a e x}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) e x}}{3 b \sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\)	\(417\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.43 \[ \int \frac {(c+d x)^2}{\sqrt {e x} \sqrt {a-b x^2}} \, dx=\frac {2 \, {\left (6 \, \sqrt {-b e} b c d {\rm weierstrassZeta}\left (\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right )\right ) - \sqrt {-b x^{2} + a} \sqrt {e x} b d^{2} - {\left (3 \, b c^{2} + a d^{2}\right )} \sqrt {-b e} {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right )\right )}}{3 \, b^{2} e} \] Input:

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

Output:

2/3*(6*sqrt(-b*e)*b*c*d*weierstrassZeta(4*a/b, 0, weierstrassPInverse(4*a/ 
b, 0, x)) - sqrt(-b*x^2 + a)*sqrt(e*x)*b*d^2 - (3*b*c^2 + a*d^2)*sqrt(-b*e 
)*weierstrassPInverse(4*a/b, 0, x))/(b^2*e)
 

Sympy [A] (verification not implemented)

Time = 4.19 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.74 \[ \int \frac {(c+d x)^2}{\sqrt {e x} \sqrt {a-b x^2}} \, dx=\frac {c^{2} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {c d x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{\sqrt {a} \sqrt {e} \Gamma \left (\frac {7}{4}\right )} + \frac {d^{2} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} \sqrt {e} \Gamma \left (\frac {9}{4}\right )} \] Input:

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

Output:

c**2*sqrt(x)*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b*x**2*exp_polar(2*I*pi) 
/a)/(2*sqrt(a)*sqrt(e)*gamma(5/4)) + c*d*x**(3/2)*gamma(3/4)*hyper((1/2, 3 
/4), (7/4,), b*x**2*exp_polar(2*I*pi)/a)/(sqrt(a)*sqrt(e)*gamma(7/4)) + d* 
*2*x**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), b*x**2*exp_polar(2*I*pi)/ 
a)/(2*sqrt(a)*sqrt(e)*gamma(9/4))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^2}{\sqrt {e x} \sqrt {a-b x^2}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, d^{2}+\left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{3}+a x}d x \right ) a \,d^{2}+3 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{3}+a x}d x \right ) b \,c^{2}+6 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{2}+a}d x \right ) b c d \right )}{3 b e} \] Input:

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

Output:

(sqrt(e)*( - 2*sqrt(x)*sqrt(a - b*x**2)*d**2 + int((sqrt(x)*sqrt(a - b*x** 
2))/(a*x - b*x**3),x)*a*d**2 + 3*int((sqrt(x)*sqrt(a - b*x**2))/(a*x - b*x 
**3),x)*b*c**2 + 6*int((sqrt(x)*sqrt(a - b*x**2))/(a - b*x**2),x)*b*c*d))/ 
(3*b*e)