\(\int \frac {A+B x+C x^2}{\sqrt {c+d x} (a-b x^2)^{3/2}} \, dx\) [211]

Optimal result

Integrand size = 32, antiderivative size = 363 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (a \left (B c-\left (A+\frac {a C}{b}\right ) d\right )+(A b c+a c C-a B d) x\right )}{a \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}+\frac {(A b c+a c C-a B d) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \sqrt {b} \left (b c^2-a d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {(A b-a C) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} b^{3/2} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

(d*x+c)^(1/2)*(a*(B*c-(A+a*C/b)*d)+(A*b*c-B*a*d+C*a*c)*x)/a/(-a*d^2+b*c^2) 
/(-b*x^2+a)^(1/2)+(A*b*c-B*a*d+C*a*c)*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*E 
llipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/ 
2)*c+a^(1/2)*d))^(1/2))/a^(1/2)/b^(1/2)/(-a*d^2+b*c^2)/((d*x+c)/(c+a^(1/2) 
*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-(A*b-C*a)*((d*x+c)/(c+a^(1/2)*d/b^(1/2 
)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2 
^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(1/2)/b^(3/2)/(d 
*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.64 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.43 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {-\left ((c+d x) \left (-a^2 C d+A b^2 c x+a b (-A d+c C x+B (c-d x))\right )\right )+\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (A b c+a c C-a B d) \left (-a+b x^2\right )-i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) (A b c+a c C-a B d) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (A b d-a C d+\sqrt {a} \sqrt {b} (-2 c C+B d)\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}}{a b \left (-b c^2+a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Input:

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

Output:

(-((c + d*x)*(-(a^2*C*d) + A*b^2*c*x + a*b*(-(A*d) + c*C*x + B*(c - d*x))) 
) + (d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(A*b*c + a*c*C - a*B*d)*(-a + b*x^ 
2) - I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(A*b*c + a*c*C - a*B*d)*Sqrt[(d*(Sq 
rt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x 
))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqr 
t[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*Sqrt[a]* 
(Sqrt[b]*c - Sqrt[a]*d)*(A*b*d - a*C*d + Sqrt[a]*Sqrt[b]*(-2*c*C + B*d))*S 
qrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x 
)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sq 
rt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/( 
d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]))/(a*b*(-(b*c^2) + a*d^2)*Sqrt[c + d*x]*S 
qrt[a - b*x^2])
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2180, 27, 600, 509, 508, 327, 512, 511, 321}

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.

\(\displaystyle \int \frac {A+B x+C x^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 2180

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

\(\displaystyle \frac {(A b-a C) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (-a B d+a c C+A b c) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (x (-a B d+a c C+A b c)+a \left (B c-d \left (\frac {a C}{b}+A\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {\sqrt {1-\frac {b x^2}{a}} (A b-a C) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (-a B d+a c C+A b c) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (x (-a B d+a c C+A b c)+a \left (B c-d \left (\frac {a C}{b}+A\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (-a B d+a c C+A b c) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} (A b-a C) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}}{2 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (x (-a B d+a c C+A b c)+a \left (B c-d \left (\frac {a C}{b}+A\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (-a B d+a c C+A b c) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} (A b-a C) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x}}}{2 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (x (-a B d+a c C+A b c)+a \left (B c-d \left (\frac {a C}{b}+A\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

Input:

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

Output:

(Sqrt[c + d*x]*(a*(B*c - (A + (a*C)/b)*d) + (A*b*c + a*c*C - a*B*d)*x))/(a 
*(b*c^2 - a*d^2)*Sqrt[a - b*x^2]) + ((2*Sqrt[a]*Sqrt[b]*(A*b*c + a*c*C - a 
*B*d)*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b] 
*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[(Sqrt[b]*(c 
 + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (2*Sqrt[a]*(A*b - a*C 
)*(b*c^2 - a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 
 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2* 
d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/(2 
*a*b*(b*c^2 - a*d^2))
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

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 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

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

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 2180

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema 
inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 
, x], x, 1]}, Simp[(-(d + e*x)^(m + 1))*(a + b*x^2)^(p + 1)*((a*(e*R - d*S) 
 + (b*d*R + a*e*S)*x)/(2*a*(p + 1)*(b*d^2 + a*e^2))), x] + Simp[1/(2*a*(p + 
 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a* 
(p + 1)*(b*d^2 + a*e^2)*Qx + b*d^2*R*(2*p + 3) - a*e*(d*S*m - e*R*(m + 2*p 
+ 3)) + e*(b*d*R + a*e*S)*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, d, e 
, m}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] &&  !(IGtQ[ 
m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(761\) vs. \(2(309)=618\).

Time = 4.45 (sec) , antiderivative size = 762, normalized size of antiderivative = 2.10

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (-\frac {\left (A b c -B a d +C a c \right ) x}{2 b \left (a \,d^{2}-b \,c^{2}\right ) a}+\frac {A b d -B b c +a C d}{2 \left (a \,d^{2}-b \,c^{2}\right ) b^{2}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 \left (-\frac {C}{b}+\frac {A b +a C}{b a}-\frac {d \left (A b d -B b c +a C d \right )}{2 b \left (a \,d^{2}-b \,c^{2}\right )}+\frac {c \left (A b c -B a d +C a c \right )}{\left (a \,d^{2}-b \,c^{2}\right ) a}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {d \left (A b c -B a d +C a c \right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) a \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\)	\(762\)
default	\(\text {Expression too large to display}\)	\(2182\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=-\frac {{\left (2 \, B a^{2} b c d - {\left (5 \, C a^{2} b - A a b^{2}\right )} c^{2} + 3 \, {\left (C a^{3} - A a^{2} b\right )} d^{2} - {\left (2 \, B a b^{2} c d - {\left (5 \, C a b^{2} - A b^{3}\right )} c^{2} + 3 \, {\left (C a^{2} b - A a b^{2}\right )} d^{2}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) - 3 \, {\left (B a^{2} b d^{2} - {\left (C a^{2} b + A a b^{2}\right )} c d - {\left (B a b^{2} d^{2} - {\left (C a b^{2} + A b^{3}\right )} c d\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (B a b^{2} c d - {\left (C a^{2} b + A a b^{2}\right )} d^{2} - {\left (B a b^{2} d^{2} - {\left (C a b^{2} + A b^{3}\right )} c d\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{3} c^{2} d - a^{3} b^{2} d^{3} - {\left (a b^{4} c^{2} d - a^{2} b^{3} d^{3}\right )} x^{2}\right )}} \] Input:

integrate((C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="frica 
s")
 

Output:

-1/3*((2*B*a^2*b*c*d - (5*C*a^2*b - A*a*b^2)*c^2 + 3*(C*a^3 - A*a^2*b)*d^2 
 - (2*B*a*b^2*c*d - (5*C*a*b^2 - A*b^3)*c^2 + 3*(C*a^2*b - A*a*b^2)*d^2)*x 
^2)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b 
*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) - 3*(B*a^2*b*d^2 - (C*a^2*b 
+ A*a*b^2)*c*d - (B*a*b^2*d^2 - (C*a*b^2 + A*b^3)*c*d)*x^2)*sqrt(-b*d)*wei 
erstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d 
^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a 
*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) - 3*(B*a*b^2*c*d - (C*a^2*b + A*a*b^2 
)*d^2 - (B*a*b^2*d^2 - (C*a*b^2 + A*b^3)*c*d)*x)*sqrt(-b*x^2 + a)*sqrt(d*x 
 + c))/(a^2*b^3*c^2*d - a^3*b^2*d^3 - (a*b^4*c^2*d - a^2*b^3*d^3)*x^2)
 

Sympy [F]

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

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

Output:

Integral((A + B*x + C*x**2)/((a - b*x**2)**(3/2)*sqrt(c + d*x)), x)
 

Maxima [F]

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

integrate((C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="maxim 
a")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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