\(\int \sqrt {c+d x} \sqrt {a-b x^2} \, dx\) [1437]

Optimal result

Integrand size = 22, antiderivative size = 357 \[ \int \sqrt {c+d x} \sqrt {a-b x^2} \, dx=-\frac {4 c \sqrt {c+d x} \sqrt {a-b x^2}}{15 d}+\frac {2 (c+d x)^{3/2} \sqrt {a-b x^2}}{5 d}-\frac {4 \sqrt {a} \left (b c^2+3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 \sqrt {b} d^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {4 \sqrt {a} c \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 \sqrt {b} d^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-4/15*c*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/d+2/5*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2 
)/d-4/15*a^(1/2)*(3*a*d^2+b*c^2)*(d*x+c)^(1/2)*(1-b*x^2/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^2/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/ 
(-b*x^2+a)^(1/2)+4/15*a^(1/2)*c*(-a*d^2+b*c^2)*(b^(1/2)*(d*x+c)/(b^(1/2)*c 
+a^(1/2)*d))^(1/2)*(1-b*x^2/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^(1/2)/d^2/ 
(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.10 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.37 \[ \int \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {2 (c+d x) (c+3 d x)}{d}+\frac {4 \left (d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2+3 a d^2\right ) \left (a-b x^2\right )+i \sqrt {b} \left (b^{3/2} c^3-\sqrt {a} b c^2 d+3 a \sqrt {b} c d^2-3 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \sqrt {b} d \left (b c^2-4 \sqrt {a} \sqrt {b} c d+3 a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{b d^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{15 \sqrt {c+d x}} \] Input:

Integrate[Sqrt[c + d*x]*Sqrt[a - b*x^2],x]
 

Output:

(Sqrt[a - b*x^2]*((2*(c + d*x)*(c + 3*d*x))/d + (4*(d^2*Sqrt[-c + (Sqrt[a] 
*d)/Sqrt[b]]*(b*c^2 + 3*a*d^2)*(a - b*x^2) + I*Sqrt[b]*(b^(3/2)*c^3 - Sqrt 
[a]*b*c^2*d + 3*a*Sqrt[b]*c*d^2 - 3*a^(3/2)*d^3)*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]], ( 
Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*Sqrt[a]*Sqrt[b]*d*(b*c 
^2 - 4*Sqrt[a]*Sqrt[b]*c*d + 3*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)*Ellip 
ticF[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)]))/(b*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b 
]]*(-a + b*x^2))))/(15*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {493, 687, 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.

\(\displaystyle \int \sqrt {a-b x^2} \sqrt {c+d x} \, dx\)

\(\Big \downarrow \) 493

\(\displaystyle \frac {2 \int \frac {(a d+b c x) \sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{5 d}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2}}{5 d}\)

\(\Big \downarrow \) 687

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (\frac {\left (3 a d^2+b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x}\right )}{5 d}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2}}{5 d}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (3 a d^2+b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x}\right )}{5 d}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2}}{5 d}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (-\frac {c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x}\right )}{5 d}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2}}{5 d}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (-\frac {c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x}\right )}{5 d}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2}}{5 d}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (-\frac {c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x}\right )}{5 d}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2}}{5 d}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x}\right )}{5 d}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2}}{5 d}\)

\(\Big \downarrow \) 321

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

Input:

Int[Sqrt[c + d*x]*Sqrt[a - b*x^2],x]
 

Output:

(2*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/(5*d) + (2*((-2*c*Sqrt[c + d*x]*Sqrt[a 
 - b*x^2])/3 + ((-2*Sqrt[a]*(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)/((Sqr 
t[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sq 
rt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*c*(b*c^2 - a*d^2)*Sqrt[(Sqrt[b]*(c 
 + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqr 
t[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sq 
rt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/3))/(5*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 493

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] + Simp[2*(p/(d*(n + 
 2*p + 1)))   Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*(a*d - b*c*x), x], x] /; 
FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && NeQ[n + 2*p + 1, 0] && ( !Rationa 
lQ[n] || LtQ[n, 1]) &&  !ILtQ[n + 2*p, 0] && IntQuadraticQ[a, 0, b, c, d, n 
, p, x]
 

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 687

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

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.37

method	result	size
risch	\(\frac {2 \left (3 d x +c \right ) \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{15 d}+\frac {2 \left (\frac {\left (3 a \,d^{2}+b \,c^{2}\right ) \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {4 a c d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{15 d \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(489\)
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {2 x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5}+\frac {2 c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{15 d}+\frac {16 a c \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{15 \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {2 a d}{5}+\frac {2 c^{2} b}{15 d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(610\)
default	\(\text {Expression too large to display}\)	\(1108\)

Input:

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

Output:

2/15*(3*d*x+c)/d*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)+2/15/d*((3*a*d^2+b*c^2)/b* 
(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/ 
d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*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(1/2*2^(1/ 
2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a 
*b)^(1/2)))^(1/2))-c/d*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^ 
(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2)))+4*a*c*d/b*(a 
*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d- 
1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d 
*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b 
/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2)))*((d*x 
+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.65 \[ \int \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\frac {2 \, {\left (2 \, {\left (b c^{3} - 9 \, a c d^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 6 \, {\left (b c^{2} d + 3 \, a d^{3}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (3 \, b d^{3} x + b c d^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{45 \, b d^{3}} \] Input:

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

Output:

2/45*(2*(b*c^3 - 9*a*c*d^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) + 6* 
(b*c^2*d + 3*a*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*b*d^3*x + b*c*d^2)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b*d^3)
 

Sympy [F]

\[ \int \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\int \sqrt {a - b x^{2}} \sqrt {c + d x}\, dx \] Input:

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

Output:

Integral(sqrt(a - b*x**2)*sqrt(c + d*x), x)
 

Maxima [F]

\[ \int \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} \sqrt {d x + c} \,d x } \] Input:

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

Output:

integrate(sqrt(-b*x^2 + a)*sqrt(d*x + c), x)
 

Giac [F]

\[ \int \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} \sqrt {d x + c} \,d x } \] Input:

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

Output:

integrate(sqrt(-b*x^2 + a)*sqrt(d*x + c), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {c+d x} \sqrt {a-b x^2} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a d +2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b c x -3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,d^{2}-\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{2}+\left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} d^{2}+3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,c^{2}}{5 b c} \] Input:

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

Output:

( - 2*sqrt(c + d*x)*sqrt(a - b*x**2)*a*d + 2*sqrt(c + d*x)*sqrt(a - b*x**2 
)*b*c*x - 3*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)*a*b*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 + 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 + 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*b*c** 
2)/(5*b*c)