\(\int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (a-b x^2)^{3/2}} \, dx\) [157]

Optimal result

Integrand size = 37, antiderivative size = 413 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (a (b B c-A b d-a C d+a c D)+b \left (c (A b+a C)-a d \left (B+\frac {a D}{b}\right )\right ) x\right )}{a b \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}+\frac {\left (A b c d+a \left (c C d+2 c^2 D-\frac {d^2 (b B+3 a D)}{b}\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \sqrt {b} d \left (b c^2-a d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {(A b d-a C d+2 a c D) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} b^{3/2} d \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.33 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.46 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {-b (c+d x) \left (A b^2 c x-a^2 (C d-c D+d D x)+a b (-A d+c C x+B (c-d x))\right )+\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (A b^2 c d+a \left (-3 a d^2 D+b \left (c C d-B d^2+2 c^2 D\right )\right )\right ) \left (-a+b x^2\right )-i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (A b^2 c d+a \left (-3 a d^2 D+b \left (c C d-B d^2+2 c^2 D\right )\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} \sqrt {b} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (A b^{3/2} d+\sqrt {a} b (-2 c C+B d)+3 a^{3/2} d D+a \sqrt {b} (-C d+2 c D)\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}}{a b^2 \left (-b c^2+a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, 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.

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

\(\Big \downarrow \) 2180

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

\(\displaystyle \frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{a b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {b \left (a \left (-\frac {d^2 (3 a D+b B)}{b}+2 c^2 D+c C d\right )+A b c d\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) (2 a c D-a C d+A b d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{a b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {b \sqrt {1-\frac {b x^2}{a}} \left (a \left (-\frac {d^2 (3 a D+b B)}{b}+2 c^2 D+c C d\right )+A b c d\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) (2 a c D-a C d+A b d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{a b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a \left (-\frac {d^2 (3 a D+b B)}{b}+2 c^2 D+c C d\right )+A b c d\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) (2 a c D-a C d+A b d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{a b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {-\frac {\left (b c^2-a d^2\right ) (2 a c D-a C d+A b d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a \left (-\frac {d^2 (3 a D+b B)}{b}+2 c^2 D+c C d\right )+A b c d\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{a b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) (2 a c D-a C d+A b d) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a \left (-\frac {d^2 (3 a D+b B)}{b}+2 c^2 D+c C d\right )+A b c d\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{a b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} (2 a c D-a C d+A b d) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a \left (-\frac {d^2 (3 a D+b B)}{b}+2 c^2 D+c C d\right )+A b c d\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a b \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{a b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} (2 a c D-a C d+A b d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a \left (-\frac {d^2 (3 a D+b B)}{b}+2 c^2 D+c C d\right )+A b c d\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a b \left (b c^2-a d^2\right )}\)

Input:

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

Output:

(Sqrt[c + d*x]*(a*(b*B*c - A*b*d - a*C*d + a*c*D) + b*(c*(A*b + a*C) - a*d 
*(B + (a*D)/b))*x))/(a*b*(b*c^2 - a*d^2)*Sqrt[a - b*x^2]) - ((-2*Sqrt[a]*S 
qrt[b]*(A*b*c*d + a*(c*C*d + 2*c^2*D - (d^2*(b*B + 3*a*D))/b))*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)])/(d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[ 
b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(A*b*d - 
a*C*d + 2*a*c*D)*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]*d*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. \(820\) vs. \(2(359)=718\).

Time = 6.46 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.99

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (-\frac {\left (A \,b^{2} c -B a b d +C a b c -a^{2} d D\right ) x}{2 a \,b^{2} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {A b d -B b c +C a d -D a c}{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 +C a}{a b}-\frac {d \left (A b d -B b c +C a d -D a c \right )}{2 b \left (a \,d^{2}-b \,c^{2}\right )}+\frac {c \left (A \,b^{2} c -B a b d +C a b c -a^{2} d D\right )}{b a \left (a \,d^{2}-b \,c^{2}\right )}\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 {2 \left (-\frac {D}{b}+\frac {d \left (A \,b^{2} c -B a b d +C a b c -a^{2} d D\right )}{2 a \left (a \,d^{2}-b \,c^{2}\right ) b}\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 )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(821\)
default	\(\text {Expression too large to display}\)	\(3231\)

Input:

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

Output:

1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((d*x+c)*(-b*x^2+a))^(1/2)*(-2*(-b*d*x-b* 
c)*(-1/2*(A*b^2*c-B*a*b*d+C*a*b*c-D*a^2*d)/a/b^2/(a*d^2-b*c^2)*x+1/2*(A*b* 
d-B*b*c+C*a*d-D*a*c)/(a*d^2-b*c^2)/b^2)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2*( 
-1/b*C+1/a*(A*b+C*a)/b-1/2/b*d*(A*b*d-B*b*c+C*a*d-D*a*c)/(a*d^2-b*c^2)+1/b 
*c*(A*b^2*c-B*a*b*d+C*a*b*c-D*a^2*d)/a/(a*d^2-b*c^2))*(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))+2*(-D/b+1/2*d*(A 
*b^2*c-B*a*b*d+C*a*b*c-D*a^2*d)/a/(a*d^2-b*c^2)/b)*(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 = 0.10 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.31 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=-\frac {{\left (2 \, D a^{2} b c^{3} + 2 \, B a^{2} b c d^{2} - {\left (5 \, C a^{2} b - A a b^{2}\right )} c^{2} d + 3 \, {\left (C a^{3} - A a^{2} b\right )} d^{3} - {\left (2 \, D a b^{2} c^{3} + 2 \, B a b^{2} c d^{2} - {\left (5 \, C a b^{2} - A b^{3}\right )} c^{2} d + 3 \, {\left (C a^{2} b - A a b^{2}\right )} d^{3}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (2 \, D a^{2} b c^{2} d + {\left (C a^{2} b + A a b^{2}\right )} c d^{2} - {\left (3 \, D a^{3} + B a^{2} b\right )} d^{3} - {\left (2 \, D a b^{2} c^{2} d + {\left (C a b^{2} + A b^{3}\right )} c d^{2} - {\left (3 \, D a^{2} b + B a b^{2}\right )} d^{3}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left ({\left (D a^{2} b + B a b^{2}\right )} c d^{2} - {\left (C a^{2} b + A a b^{2}\right )} d^{3} + {\left ({\left (C a b^{2} + A b^{3}\right )} c d^{2} - {\left (D a^{2} b + B a b^{2}\right )} d^{3}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{3} c^{2} d^{2} - a^{3} b^{2} d^{4} - {\left (a b^{4} c^{2} d^{2} - a^{2} b^{3} d^{4}\right )} x^{2}\right )}} \] Input:

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

Output:

-1/3*((2*D*a^2*b*c^3 + 2*B*a^2*b*c*d^2 - (5*C*a^2*b - A*a*b^2)*c^2*d + 3*( 
C*a^3 - A*a^2*b)*d^3 - (2*D*a*b^2*c^3 + 2*B*a*b^2*c*d^2 - (5*C*a*b^2 - A*b 
^3)*c^2*d + 3*(C*a^2*b - A*a*b^2)*d^3)*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*(2*D*a^2*b*c^2*d + (C*a^2*b + A*a*b^2)*c*d^2 - (3*D*a^3 + 
B*a^2*b)*d^3 - (2*D*a*b^2*c^2*d + (C*a*b^2 + A*b^3)*c*d^2 - (3*D*a^2*b + B 
*a*b^2)*d^3)*x^2)*sqrt(-b*d)*weierstrassZeta(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*( 
(D*a^2*b + B*a*b^2)*c*d^2 - (C*a^2*b + A*a*b^2)*d^3 + ((C*a*b^2 + A*b^3)*c 
*d^2 - (D*a^2*b + B*a*b^2)*d^3)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a^2*b^ 
3*c^2*d^2 - a^3*b^2*d^4 - (a*b^4*c^2*d^2 - a^2*b^3*d^4)*x^2)
 

Sympy [F]

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

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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