\(\int \frac {\sqrt {c+d x} (A+B x+C x^2+D x^3)}{(a-b x^2)^{3/2}} \, dx\) [156]

Optimal result

Integrand size = 37, antiderivative size = 399 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\left (a \left (B+\frac {a D}{b}\right )+(A b+a C) x\right ) \sqrt {c+d x}}{a b \sqrt {a-b x^2}}+\frac {2 D \sqrt {c+d x} \sqrt {a-b x^2}}{3 b^2}+\frac {(3 A b d+9 a C d+2 a c D) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {a} b^{3/2} d \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\left (3 A b^2 c d-a \left (5 a d^2 D-b \left (3 c C d-3 B d^2+2 c^2 D\right )\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {a} b^{5/2} d \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

(a*(B+a*D/b)+(A*b+C*a)*x)*(d*x+c)^(1/2)/a/b/(-b*x^2+a)^(1/2)+2/3*D*(d*x+c) 
^(1/2)*(-b*x^2+a)^(1/2)/b^2+1/3*(3*A*b*d+9*C*a*d+2*D*a*c)*(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^(3/2)/d/((d*x+c)/(c+ 
a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-1/3*(3*A*b^2*c*d-a*(5*a*d^2*D-b 
*(-3*B*d^2+3*C*c*d+2*D*c^2)))*((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^(5/2)/d/(d*x+c)^(1/2)/(- 
b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.31 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (5 a^2 D+3 A b^2 x+a b (3 B+x (3 C-2 D x))\right )}{a b^2 \left (a-b x^2\right )}+\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (3 A b d+9 a C d+2 a c D) \left (-a+b x^2\right )-i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) (3 A b d+9 a C d+2 a c D) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} d \left (3 A b^{3/2} d-3 \sqrt {a} b (2 c C+B d)-5 a^{3/2} d D+a \sqrt {b} (9 C d+2 c D)\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{a b^2 d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{3 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*(((c + d*x)*(5*a^2*D + 3*A*b^2*x + a*b*(3*B + x*(3*C - 2* 
D*x))))/(a*b^2*(a - b*x^2)) + (d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(3*A*b*d 
 + 9*a*C*d + 2*a*c*D)*(-a + b*x^2) - I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(3* 
A*b*d + 9*a*C*d + 2*a*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)*EllipticE[I*ArcS 
inh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d) 
/(Sqrt[b]*c - Sqrt[a]*d)] - I*Sqrt[a]*d*(3*A*b^(3/2)*d - 3*Sqrt[a]*b*(2*c* 
C + B*d) - 5*a^(3/2)*d*D + a*Sqrt[b]*(9*C*d + 2*c*D))*Sqrt[(d*(Sqrt[a]/Sqr 
t[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)])/(a*b^2*d^2*Sqrt[-c + 
 (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(3*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {2176, 27, 2185, 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.

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

\(\Big \downarrow \) 2176

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {-\frac {2 \int -\frac {d^2 (a (6 b c C+3 b B d+5 a d D)+b (3 A b d+9 a C d+2 a c D) x)}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}-\frac {4 a D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}}{2 a b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\int \frac {a (6 b c C+3 b B d+5 a d D)+b (3 A b d+9 a C d+2 a c D) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {4 a D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}}{2 a b}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {b (2 a c D+9 a C d+3 A b d) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (3 A b^2 c d-a \left (5 a d^2 D-b \left (-3 B d^2+2 c^2 D+3 c C d\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}-\frac {4 a D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}}{2 a b}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {b \sqrt {1-\frac {b x^2}{a}} (2 a c D+9 a C d+3 A b d) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (3 A b^2 c d-a \left (5 a d^2 D-b \left (-3 B d^2+2 c^2 D+3 c C d\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}-\frac {4 a D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}}{2 a b}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {-\frac {\left (3 A b^2 c d-a \left (5 a d^2 D-b \left (-3 B d^2+2 c^2 D+3 c C d\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (2 a c D+9 a C d+3 A b d) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {4 a D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}}{2 a b}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {-\frac {\left (3 A b^2 c d-a \left (5 a d^2 D-b \left (-3 B d^2+2 c^2 D+3 c C d\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (2 a c D+9 a C d+3 A b d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {4 a D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}}{2 a b}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (3 A b^2 c d-a \left (5 a d^2 D-b \left (-3 B d^2+2 c^2 D+3 c C d\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (2 a c D+9 a C d+3 A b d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {4 a D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}}{2 a b}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (3 A b^2 c d-a \left (5 a d^2 D-b \left (-3 B d^2+2 c^2 D+3 c C d\right )\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (2 a c D+9 a C d+3 A b d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {4 a D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}}{2 a b}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 A b^2 c d-a \left (5 a d^2 D-b \left (-3 B d^2+2 c^2 D+3 c C d\right )\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (2 a c D+9 a C d+3 A b d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {4 a D \sqrt {a-b x^2} \sqrt {c+d x}}{3 b}}{2 a b}\)

Input:

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

Output:

((a*(B + (a*D)/b) + (A*b + a*C)*x)*Sqrt[c + d*x])/(a*b*Sqrt[a - b*x^2]) - 
((-4*a*D*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(3*b) + ((-2*Sqrt[a]*Sqrt[b]*(3*A* 
b*d + 9*a*C*d + 2*a*c*D)*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSi 
n[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]*(3*A*b^2*c*d - a*(5*a*d^2*D - b*(3*c*C*d - 3*B*d^2 + 2*c^2*D))) 
*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*Ell 
ipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/S 
qrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/(3*b))/(2*a*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 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 2176

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*(a + b*x^2)^(p + 1)*((a*S - b*R*x)/(2*a*b*(p 
 + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^(p 
 + 1)*ExpandToSum[2*a*b*(p + 1)*(d + e*x)*Qx - a*e*S*m + b*d*R*(2*p + 3) + 
b*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] &&  !(IGtQ[m, 0] && R 
ationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(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. \(752\) vs. \(2(335)=670\).

Time = 4.40 (sec) , antiderivative size = 753, normalized size of antiderivative = 1.89

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 +C a \right ) x}{2 a \,b^{2}}+\frac {B b +D a}{2 b^{3}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 D \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b^{2}}+\frac {2 \left (-\frac {B b d +b c C +D a d}{b^{2}}+\frac {A \,b^{2} c +B a b d +C a b c +a^{2} d D}{b^{2} a}-\frac {d \left (B b +D a \right )}{2 b^{2}}-\frac {c \left (A b +C a \right )}{b a}-\frac {D a d}{3 b^{2}}\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 {C d +D c}{b}-\frac {\left (A b +C a \right ) d}{2 a b}+\frac {2 D c}{3 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}}\)	\(753\)
default	\(\text {Expression too large to display}\)	\(2470\)

Input:

int((d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A)/(-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*(A*b+C*a)/b^2*x+1/2*(B*b+D*a)/b^3)/((x^2-a/b)*(-b*d*x-b*c))^(1/2 
)+2/3*D/b^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-(B*b*d+C*b*c+D*a*d)/b^2 
+1/b^2*(A*b^2*c+B*a*b*d+C*a*b*c+D*a^2*d)/a-1/2/b^2*d*(B*b+D*a)-1/b*c/a*(A* 
b+C*a)-1/3*D/b^2*a*d)*(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)*Ell 
ipticF(((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*(-(C*d+D*c)/b-1/2*(A*b+C*a)*d/a/b+2/3*D/b*c)*( 
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))*El 
lipticE(((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 = 427, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {{\left (2 \, D a^{2} b c^{2} - 3 \, {\left (3 \, C a^{2} b - A a b^{2}\right )} c d - 3 \, {\left (5 \, D a^{3} + 3 \, B a^{2} b\right )} d^{2} - {\left (2 \, D a b^{2} c^{2} - 3 \, {\left (3 \, C a b^{2} - A b^{3}\right )} c d - 3 \, {\left (5 \, D a^{2} b + 3 \, B a b^{2}\right )} d^{2}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (2 \, D a^{2} b c d + 3 \, {\left (3 \, C a^{2} b + A a b^{2}\right )} d^{2} - {\left (2 \, D a b^{2} c d + 3 \, {\left (3 \, C a b^{2} + A b^{3}\right )} d^{2}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (2 \, D a b^{2} d^{2} x^{2} - 3 \, {\left (C a b^{2} + A b^{3}\right )} d^{2} x - {\left (5 \, D a^{2} b + 3 \, B a b^{2}\right )} d^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{9 \, {\left (a b^{4} d^{2} x^{2} - a^{2} b^{3} d^{2}\right )}} \] Input:

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

Output:

1/9*((2*D*a^2*b*c^2 - 3*(3*C*a^2*b - A*a*b^2)*c*d - 3*(5*D*a^3 + 3*B*a^2*b 
)*d^2 - (2*D*a*b^2*c^2 - 3*(3*C*a*b^2 - A*b^3)*c*d - 3*(5*D*a^2*b + 3*B*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*(2*D*a^2*b*c* 
d + 3*(3*C*a^2*b + A*a*b^2)*d^2 - (2*D*a*b^2*c*d + 3*(3*C*a*b^2 + A*b^3)*d 
^2)*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*(2*D*a*b^2 
*d^2*x^2 - 3*(C*a*b^2 + A*b^3)*d^2*x - (5*D*a^2*b + 3*B*a*b^2)*d^2)*sqrt(- 
b*x^2 + a)*sqrt(d*x + c))/(a*b^4*d^2*x^2 - a^2*b^3*d^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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