\(\int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx\) [150]

Optimal result

Integrand size = 37, antiderivative size = 410 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 (5 C d-7 c D) \sqrt {c+d x} \sqrt {a-b x^2}}{15 b d^2}-\frac {2 D (c+d x)^{3/2} \sqrt {a-b x^2}}{5 b d^2}-\frac {2 \sqrt {a} \left (9 a d^2 D-b \left (10 c C d-15 B d^2-8 c^2 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 b^{3/2} d^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (a d^2 (5 C d-7 c D)+b \left (10 c^2 C d-15 B c d^2+15 A d^3-8 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 b^{3/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-2/15*(5*C*d-7*D*c)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^2-2/5*D*(d*x+c)^(3/ 
2)*(-b*x^2+a)^(1/2)/b/d^2-2/15*a^(1/2)*(9*a*d^2*D-b*(-15*B*d^2+10*C*c*d-8* 
D*c^2))*(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))/b^(3/2 
)/d^3/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-2/15*a^(1/2)* 
(a*d^2*(5*C*d-7*D*c)+b*(15*A*d^3-15*B*c*d^2+10*C*c^2*d-8*D*c^3))*((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)) 
/b^(3/2)/d^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.80 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.33 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (-9 a d^2 D-b \left (-10 c C d+15 B d^2+8 c^2 D\right )+b (c+d x) (-5 C d+4 c D-3 d D x)+\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (9 a d^2 D+b \left (-10 c C d+15 B d^2+8 c^2 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}-\frac {i \sqrt {b} \left (15 A b^{3/2} d^2-9 a^{3/2} d^2 D+a \sqrt {b} d (5 C d+2 c D)+\sqrt {a} b \left (10 c C d-15 B d^2-8 c^2 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{15 b^2 d^2 \sqrt {c+d x}} \] Input:

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

Output:

(2*Sqrt[a - b*x^2]*(-9*a*d^2*D - b*(-10*c*C*d + 15*B*d^2 + 8*c^2*D) + b*(c 
 + d*x)*(-5*C*d + 4*c*D - 3*d*D*x) + (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(9 
*a*d^2*D + b*(-10*c*C*d + 15*B*d^2 + 8*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)/Sqrt[b]]/Sqrt[c + d*x]], (Sq 
rt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d^2*Sqrt[-c + (Sqrt[a]*d)/ 
Sqrt[b]]*(-a + b*x^2)) - (I*Sqrt[b]*(15*A*b^(3/2)*d^2 - 9*a^(3/2)*d^2*D + 
a*Sqrt[b]*d*(5*C*d + 2*c*D) + Sqrt[a]*b*(10*c*C*d - 15*B*d^2 - 8*c^2*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*x^2))))/(15*b^2*d^2*Sqrt[c + d*x] 
)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {2185, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle -\frac {2 \int -\frac {b (5 C d-7 c D) x^2 d^2+(5 A b d+3 a c D) d^2+\left (-2 b D c^2+5 b B d^2+3 a d^2 D\right ) x d}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {b (5 C d-7 c D) x^2 d^2+(5 A b d+3 a c D) d^2+\left (-2 b D c^2+5 b B d^2+3 a d^2 D\right ) x d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {-\frac {2 \int -\frac {b d^3 \left (d (15 A b d+5 a C d+2 a c D)+\left (9 a d^2 D-b \left (-8 D c^2+10 C d c-15 B d^2\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 C d-7 c D)}{5 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} d \int \frac {d (15 A b d+5 a C d+2 a c D)+\left (9 a d^2 D-b \left (-8 D c^2+10 C d c-15 B d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 C d-7 c D)}{5 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {1}{3} d \left (\frac {\left (a d^2 (5 C d-7 c D)+b \left (15 A d^3-15 B c d^2-8 c^3 D+10 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {\left (9 a d^2 D-b \left (-15 B d^2-8 c^2 D+10 c C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 C d-7 c D)}{5 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {1}{3} d \left (\frac {\left (a d^2 (5 C d-7 c D)+b \left (15 A d^3-15 B c d^2-8 c^3 D+10 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {\sqrt {1-\frac {b x^2}{a}} \left (9 a d^2 D-b \left (-15 B d^2-8 c^2 D+10 c C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 C d-7 c D)}{5 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {1}{3} d \left (\frac {\left (a d^2 (5 C d-7 c D)+b \left (15 A d^3-15 B c d^2-8 c^3 D+10 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a d^2 D-b \left (-15 B d^2-8 c^2 D+10 c C d\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 C d-7 c D)}{5 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {1}{3} d \left (\frac {\left (a d^2 (5 C d-7 c D)+b \left (15 A d^3-15 B c d^2-8 c^3 D+10 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (9 a d^2 D-b \left (-15 B d^2-8 c^2 D+10 c C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 C d-7 c D)}{5 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {1}{3} d \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (a d^2 (5 C d-7 c D)+b \left (15 A d^3-15 B c d^2-8 c^3 D+10 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (9 a d^2 D-b \left (-15 B d^2-8 c^2 D+10 c C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 C d-7 c D)}{5 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {1}{3} d \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (a d^2 (5 C d-7 c D)+b \left (15 A d^3-15 B c d^2-8 c^3 D+10 c^2 C d\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (9 a d^2 D-b \left (-15 B d^2-8 c^2 D+10 c C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 C d-7 c D)}{5 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {1}{3} d \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (5 C d-7 c D)+b \left (15 A d^3-15 B c d^2-8 c^3 D+10 c^2 C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (9 a d^2 D-b \left (-15 B d^2-8 c^2 D+10 c C d\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 C d-7 c D)}{5 b d^3}-\frac {2 D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\)

Input:

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

Output:

(-2*D*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/(5*b*d^2) + ((-2*d*(5*C*d - 7*c*D)* 
Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 + (d*((-2*Sqrt[a]*(9*a*d^2*D - b*(10*c*C* 
d - 15*B*d^2 - 8*c^2*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)] 
)/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b* 
x^2]) - (2*Sqrt[a]*(a*d^2*(5*C*d - 7*c*D) + b*(10*c^2*C*d - 15*B*c*d^2 + 1 
5*A*d^3 - 8*c^3*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]) 
))/3)/(5*b*d^3)
 

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 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 [A] (verified)

Time = 3.96 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.62

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 D x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b d}-\frac {2 \left (C -\frac {4 D c}{5 d}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (A +\frac {2 D a c}{5 b d}+\frac {\left (C -\frac {4 D c}{5 d}\right ) a}{3 b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (B +\frac {3 a D}{5 b}-\frac {2 \left (C -\frac {4 D c}{5 d}\right ) c}{3 d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(664\)
default	\(\text {Expression too large to display}\)	\(2734\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),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)*(-2/5*D/b/d*x* 
(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(C-4/5*D/d*c)/b/d*(-b*d*x^3-b*c*x^2 
+a*d*x+a*c)^(1/2)+2*(A+2/5*D/b/d*a*c+1/3*(C-4/5*D/d*c)/b*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))+2*(B+3/5*a 
*D/b-2/3*(C-4/5*D/d*c)/d*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))*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)*El 
lipticF(((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 = 0.08 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 \, {\left ({\left (8 \, D b c^{3} - 10 \, C b c^{2} d + 3 \, {\left (D a + 5 \, B b\right )} c d^{2} - 15 \, {\left (C a + 3 \, A b\right )} d^{3}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (8 \, D b c^{2} d - 10 \, C b c d^{2} + 3 \, {\left (3 \, D a + 5 \, B b\right )} d^{3}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (3 \, D b d^{3} x - 4 \, D b c d^{2} + 5 \, C b d^{3}\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{45 \, b^{2} d^{4}} \] Input:

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

Output:

2/45*((8*D*b*c^3 - 10*C*b*c^2*d + 3*(D*a + 5*B*b)*c*d^2 - 15*(C*a + 3*A*b) 
*d^3)*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*(8*D*b*c^2*d - 10*C*b* 
c*d^2 + 3*(3*D*a + 5*B*b)*d^3)*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*(3*D*b*d^3*x - 4*D*b*c*d^2 + 5*C*b*d^3)*sqrt(-b*x^2 + a)*sqrt( 
d*x + c))/(b^2*d^4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {-6 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a d -10 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2}-4 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b c x -9 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,d^{2}-15 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{3} d +2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{2}+3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} d^{2}+10 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c +5 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} d +4 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,c^{2}}{10 b^{2} c} \] Input:

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

Output:

( - 6*sqrt(c + d*x)*sqrt(a - b*x**2)*a*d - 10*sqrt(c + d*x)*sqrt(a - b*x** 
2)*b**2 - 4*sqrt(c + d*x)*sqrt(a - b*x**2)*b*c*x - 9*int((sqrt(c + d*x)*sq 
rt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b*d**2 - 15* 
int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x* 
*3),x)*b**3*d + 2*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - 
 b*c*x**2 - b*d*x**3),x)*b**2*c**2 + 3*int((sqrt(c + d*x)*sqrt(a - b*x**2) 
)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*d**2 + 10*int((sqrt(c + d*x) 
*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b**2*c + 5*int 
((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a 
*b**2*d + 4*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - 
 b*d*x**3),x)*a*b*c**2)/(10*b**2*c)