\(\int \frac {(c+d x)^2}{(e x)^{3/2} \sqrt {a-b x^2}} \, dx\) [1382]

Optimal result

Integrand size = 27, antiderivative size = 209 \[ \int \frac {(c+d x)^2}{(e x)^{3/2} \sqrt {a-b x^2}} \, dx=-\frac {2 c^2 \sqrt {a-b x^2}}{a e \sqrt {e x}}-\frac {2 \left (b c^2-a d^2\right ) \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{\sqrt [4]{a} b^{3/4} e^{3/2} \sqrt {a-b x^2}}+\frac {2 \left (b c^2+2 \sqrt {a} \sqrt {b} c d-a d^2\right ) \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{a} b^{3/4} e^{3/2} \sqrt {a-b x^2}} \] Output:

-2*c^2*(-b*x^2+a)^(1/2)/a/e/(e*x)^(1/2)-2*(-a*d^2+b*c^2)*(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)/b^(3/4)/e^(3/2)/ 
(-b*x^2+a)^(1/2)+2*(b*c^2+2*a^(1/2)*b^(1/2)*c*d-a*d^2)*(1-b*x^2/a)^(1/2)*E 
llipticF(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I)/a^(1/4)/b^(3/4)/e^(3/2)/(- 
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 = 109, normalized size of antiderivative = 0.52 \[ \int \frac {(c+d x)^2}{(e x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {2 x \sqrt {1-\frac {b x^2}{a}} \left (-3 c^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\frac {b x^2}{a}\right )+d x \left (6 c \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^2}{a}\right )+d x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^2}{a}\right )\right )\right )}{3 (e x)^{3/2} \sqrt {a-b x^2}} \] Input:

Integrate[(c + d*x)^2/((e*x)^(3/2)*Sqrt[a - b*x^2]),x]
 

Output:

(2*x*Sqrt[1 - (b*x^2)/a]*(-3*c^2*Hypergeometric2F1[-1/4, 1/2, 3/4, (b*x^2) 
/a] + d*x*(6*c*Hypergeometric2F1[1/4, 1/2, 5/4, (b*x^2)/a] + d*x*Hypergeom 
etric2F1[1/2, 3/4, 7/4, (b*x^2)/a])))/(3*(e*x)^(3/2)*Sqrt[a - b*x^2])
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {559, 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.

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

\(\Big \downarrow \) 559

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 553

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 556

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

\(\Big \downarrow \) 555

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

\(\Big \downarrow \) 1513

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

Int[(c + d*x)^2/((e*x)^(3/2)*Sqrt[a - b*x^2]),x]
 

Output:

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

Time = 0.99 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.58

method	result	size
risch	\(-\frac {2 c^{2} \sqrt {-b \,x^{2}+a}}{a e \sqrt {e x}}+\frac {\left (\frac {\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {-b e \,x^{3}+a e x}}+\frac {2 a c d \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) e x}}{a e \sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\)	\(330\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (-\frac {2 \left (-b e \,x^{2}+a e \right ) c^{2}}{e^{2} a \sqrt {x \left (-b e \,x^{2}+a e \right )}}+\frac {2 c d \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{e b \sqrt {-b e \,x^{3}+a e x}}+\frac {\left (\frac {d^{2}}{e}-\frac {b \,c^{2}}{a e}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {-b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\)	\(344\)
default	\(\frac {-2 \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} d^{2}+2 \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b \,c^{2}+2 \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {a b}\, \sqrt {2}\, a c d +\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} d^{2}-\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b \,c^{2}+2 b^{2} c^{2} x^{2}-2 a b \,c^{2}}{\sqrt {-b \,x^{2}+a}\, b e \sqrt {e x}\, a}\)	\(444\)

Input:

int((d*x+c)^2/(e*x)^(3/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.44 \[ \int \frac {(c+d x)^2}{(e x)^{3/2} \sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {-b e} a c d x {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right ) + \sqrt {-b x^{2} + a} \sqrt {e x} b c^{2} + {\left (b c^{2} - a d^{2}\right )} \sqrt {-b e} x {\rm weierstrassZeta}\left (\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right )\right )\right )}}{a b e^{2} x} \] Input:

integrate((d*x+c)^2/(e*x)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 4.86 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.73 \[ \int \frac {(c+d x)^2}{(e x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {c^{2} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {c d \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{\sqrt {a} e^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {d^{2} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \] Input:

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

Output:

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

Maxima [F]

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

integrate((d*x+c)^2/(e*x)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((d*x+c)^2/(e*x)^(3/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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