\(\int \frac {\sqrt {a-b x^2} (A+B x+C x^2)}{\sqrt {c+d x}} \, dx\) [170]

Optimal result

Integrand size = 32, antiderivative size = 468 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\frac {2 \left (35 A b+5 a C+\frac {4 b c (6 c C-7 B d)}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{105 b d}-\frac {2 (6 c C-7 B d) x \sqrt {c+d x} \sqrt {a-b x^2}}{35 d^2}-\frac {2 C \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{7 b d}+\frac {4 \sqrt {a} \left (a d^2 (13 c C-21 B d)-b c \left (24 c^2 C-28 B c d+35 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 \sqrt {b} d^4 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {4 \sqrt {a} \left (b c^2-a d^2\right ) \left (5 a C d^2+b \left (24 c^2 C-28 B c d+35 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 b^{3/2} d^4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

2/105*(35*A*b+5*a*C+4*b*c*(-7*B*d+6*C*c)/d^2)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/ 
2)/b/d-2/35*(-7*B*d+6*C*c)*x*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/d^2-2/7*C*(d*x 
+c)^(1/2)*(-b*x^2+a)^(3/2)/b/d+4/105*a^(1/2)*(a*d^2*(-21*B*d+13*C*c)-b*c*( 
35*A*d^2-28*B*c*d+24*C*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^(1/2)/d^4/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a 
)^(1/2)+4/105*a^(1/2)*(-a*d^2+b*c^2)*(5*a*C*d^2+b*(35*A*d^2-28*B*c*d+24*C* 
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))/b^(3/2)/d^4/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 27.06 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\frac {2 \sqrt {a-b x^2} \left ((c+d x) \left (-10 a C d^2+b \left (24 c^2 C-2 c d (14 B+9 C x)+d^2 (35 A+3 x (7 B+5 C x))\right )\right )-\frac {2 \left (d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a d^2 (-13 c C+21 B d)+b c \left (24 c^2 C-28 B c d+35 A d^2\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2 (-13 c C+21 B d)+b c \left (24 c^2 C-28 B c d+35 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (5 a C d^2+3 \sqrt {a} \sqrt {b} d (6 c C-7 B d)+b \left (24 c^2 C-28 B c d+35 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{105 b d^3 \sqrt {c+d x}} \] Input:

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

Output:

(2*Sqrt[a - b*x^2]*((c + d*x)*(-10*a*C*d^2 + b*(24*c^2*C - 2*c*d*(14*B + 9 
*C*x) + d^2*(35*A + 3*x*(7*B + 5*C*x)))) - (2*(d^2*Sqrt[-c + (Sqrt[a]*d)/S 
qrt[b]]*(a*d^2*(-13*c*C + 21*B*d) + b*c*(24*c^2*C - 28*B*c*d + 35*A*d^2))* 
(a - b*x^2) + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(a*d^2*(-13*c*C + 21*B*d) 
+ b*c*(24*c^2*C - 28*B*c*d + 35*A*d^2))*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)*Ell 
ipticE[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)] + I*Sqrt[a]*d*(Sqrt[b]*c - Sqrt[a]* 
d)*(5*a*C*d^2 + 3*Sqrt[a]*Sqrt[b]*d*(6*c*C - 7*B*d) + b*(24*c^2*C - 28*B*c 
*d + 35*A*d^2))*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 + (Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2))))/(105*b 
*d^3*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 457, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2185, 27, 682, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle -\frac {2 \int -\frac {d ((7 A b+a C) d-b (6 c C-7 B d) x) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{7 b d^2}-\frac {2 C \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {((7 A b+a C) d-b (6 c C-7 B d) x) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{7 b d}-\frac {2 C \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b d}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 d^2 (a C+7 A b)-3 b d x (6 c C-7 B d)+4 b c (6 c C-7 B d)\right )}{15 d^2}-\frac {4 \int -\frac {b \left (a d \left (5 (7 A b+a C) d^2+b c (6 c C-7 B d)\right )+b \left (5 c (7 A b+a C) d^2+(6 c C-7 B d) \left (4 b c^2-3 a d^2\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}}{7 b d}-\frac {2 C \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \int \frac {a d \left (5 (7 A b+a C) d^2+b c (6 c C-7 B d)\right )+b \left (5 c (7 A b+a C) d^2+(6 c C-7 B d) \left (4 b c^2-3 a d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 d^2 (a C+7 A b)-3 b d x (6 c C-7 B d)+4 b c (6 c C-7 B d)\right )}{15 d^2}}{7 b d}-\frac {2 C \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b d}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {2 \left (\frac {b \left (5 c d^2 (a C+7 A b)+\left (4 b c^2-3 a d^2\right ) (6 c C-7 B d)\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (5 a C d^2+b \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 d^2 (a C+7 A b)-3 b d x (6 c C-7 B d)+4 b c (6 c C-7 B d)\right )}{15 d^2}}{7 b d}-\frac {2 C \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b d}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {2 \left (\frac {b \sqrt {1-\frac {b x^2}{a}} \left (5 c d^2 (a C+7 A b)+\left (4 b c^2-3 a d^2\right ) (6 c C-7 B d)\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (5 a C d^2+b \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 d^2 (a C+7 A b)-3 b d x (6 c C-7 B d)+4 b c (6 c C-7 B d)\right )}{15 d^2}}{7 b d}-\frac {2 C \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b d}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (5 a C d^2+b \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (5 c d^2 (a C+7 A b)+\left (4 b c^2-3 a d^2\right ) (6 c C-7 B d)\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 d^2 (a C+7 A b)-3 b d x (6 c C-7 B d)+4 b c (6 c C-7 B d)\right )}{15 d^2}}{7 b d}-\frac {2 C \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b d}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (5 a C d^2+b \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (5 c d^2 (a C+7 A b)+\left (4 b c^2-3 a d^2\right ) (6 c C-7 B d)\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 d^2 (a C+7 A b)-3 b d x (6 c C-7 B d)+4 b c (6 c C-7 B d)\right )}{15 d^2}}{7 b d}-\frac {2 C \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b d}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {2 \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (5 a C d^2+b \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (5 c d^2 (a C+7 A b)+\left (4 b c^2-3 a d^2\right ) (6 c C-7 B d)\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 d^2 (a C+7 A b)-3 b d x (6 c C-7 B d)+4 b c (6 c C-7 B d)\right )}{15 d^2}}{7 b d}-\frac {2 C \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b d}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (5 a C d^2+b \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (5 c d^2 (a C+7 A b)+\left (4 b c^2-3 a d^2\right ) (6 c C-7 B d)\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 d^2 (a C+7 A b)-3 b d x (6 c C-7 B d)+4 b c (6 c C-7 B d)\right )}{15 d^2}}{7 b d}-\frac {2 C \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b d}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (5 a C d^2+b \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (5 c d^2 (a C+7 A b)+\left (4 b c^2-3 a d^2\right ) (6 c C-7 B d)\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (5 d^2 (a C+7 A b)-3 b d x (6 c C-7 B d)+4 b c (6 c C-7 B d)\right )}{15 d^2}}{7 b d}-\frac {2 C \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}{7 b d}\)

Input:

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

Output:

(-2*C*Sqrt[c + d*x]*(a - b*x^2)^(3/2))/(7*b*d) + ((2*Sqrt[c + d*x]*(5*(7*A 
*b + a*C)*d^2 + 4*b*c*(6*c*C - 7*B*d) - 3*b*d*(6*c*C - 7*B*d)*x)*Sqrt[a - 
b*x^2])/(15*d^2) + (2*((-2*Sqrt[a]*Sqrt[b]*(5*c*(7*A*b + a*C)*d^2 + (6*c*C 
 - 7*B*d)*(4*b*c^2 - 3*a*d^2))*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)*(5*a*C*d^2 + b*(24*c^2*C - 28*B*c*d + 35* 
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]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(15*d^2)) 
/(7*b*d)
 

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 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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. \(801\) vs. \(2(396)=792\).

Time = 3.24 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.71

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (\frac {2 C \,x^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{7 d}-\frac {2 \left (-B b +\frac {6 C b c}{7 d}\right ) x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b d}-\frac {2 \left (-A b +\frac {2 a C}{7}-\frac {4 \left (-B b +\frac {6 C b c}{7 d}\right ) 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 a +\frac {2 \left (-B b +\frac {6 C b c}{7 d}\right ) a c}{5 b d}+\frac {\left (-A b +\frac {2 a C}{7}-\frac {4 \left (-B b +\frac {6 C b c}{7 d}\right ) 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 a -\frac {4 C a c}{7 d}+\frac {3 \left (-B b +\frac {6 C b c}{7 d}\right ) a}{5 b}-\frac {2 \left (-A b +\frac {2 a C}{7}-\frac {4 \left (-B b +\frac {6 C b c}{7 d}\right ) 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 {-b \,x^{2}+a}\, \sqrt {d x +c}}\)	\(802\)
risch	\(\text {Expression too large to display}\)	\(1067\)
default	\(\text {Expression too large to display}\)	\(3057\)

Input:

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

Output:

((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(2/7*C/d*x^2*(-b 
*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(-B*b+6/7*C/d*b*c)/b/d*x*(-b*d*x^3-b*c 
*x^2+a*d*x+a*c)^(1/2)-2/3*(-A*b+2/7*a*C-4/5*(-B*b+6/7*C/d*b*c)/d*c)/b/d*(- 
b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(A*a+2/5*(-B*b+6/7*C/d*b*c)/b/d*a*c+1/3 
*(-A*b+2/7*a*C-4/5*(-B*b+6/7*C/d*b*c)/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*a-4/7*C/d*a*c+3/5 
*(-B*b+6/7*C/d*b*c)/b*a-2/3*(-A*b+2/7*a*C-4/5*(-B*b+6/7*C/d*b*c)/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)*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 = 0.08 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (2 \, {\left (24 \, C b^{2} c^{4} - 28 \, B b^{2} c^{3} d + 42 \, B a b c d^{3} - {\left (31 \, C a b - 35 \, A b^{2}\right )} c^{2} d^{2} - 15 \, {\left (C a^{2} + 7 \, A a b\right )} d^{4}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 6 \, {\left (24 \, C b^{2} c^{3} d - 28 \, B b^{2} c^{2} d^{2} + 21 \, B a b d^{4} - {\left (13 \, C a b - 35 \, A b^{2}\right )} c d^{3}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (15 \, C b^{2} d^{4} x^{2} + 24 \, C b^{2} c^{2} d^{2} - 28 \, B b^{2} c d^{3} - 5 \, {\left (2 \, C a b - 7 \, A b^{2}\right )} d^{4} - 3 \, {\left (6 \, C b^{2} c d^{3} - 7 \, B b^{2} d^{4}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{315 \, b^{2} d^{5}} \] Input:

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

Output:

2/315*(2*(24*C*b^2*c^4 - 28*B*b^2*c^3*d + 42*B*a*b*c*d^3 - (31*C*a*b - 35* 
A*b^2)*c^2*d^2 - 15*(C*a^2 + 7*A*a*b)*d^4)*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) + 6*(24*C*b^2*c^3*d - 28*B*b^2*c^2*d^2 + 21*B*a*b*d^4 - (13*C*a 
*b - 35*A*b^2)*c*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*(15*C*b^2*d^4*x^2 + 24*C*b^2*c^2*d^2 - 28*B*b^2*c*d^3 - 5*(2*C*a*b - 7* 
A*b^2)*d^4 - 3*(6*C*b^2*c*d^3 - 7*B*b^2*d^4)*x)*sqrt(-b*x^2 + a)*sqrt(d*x 
+ c))/(b^2*d^5)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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