\(\int (c+d \sqrt {a+b x^2})^{3/2} \, dx\) [262]

Optimal result

Integrand size = 19, antiderivative size = 335 \[ \int \left (c+d \sqrt {a+b x^2}\right )^{3/2} \, dx=\frac {2}{5} c x \sqrt {c+d \sqrt {a+b x^2}}+\frac {2}{5} x \left (c+d \sqrt {a+b x^2}\right )^{3/2}-\frac {2 \sqrt {a} \left (c^2+3 a d^2\right ) \sqrt {-\frac {b x^2}{a}} \sqrt {c+d \sqrt {a+b x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{5 b d x \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}}}+\frac {2 \sqrt {a} c \left (c^2-a d^2\right ) \sqrt {-\frac {b x^2}{a}} \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{5 b d x \sqrt {c+d \sqrt {a+b x^2}}} \] Output:

2/5*c*x*(c+d*(b*x^2+a)^(1/2))^(1/2)+2/5*x*(c+d*(b*x^2+a)^(1/2))^(3/2)-2/5* 
a^(1/2)*(3*a*d^2+c^2)*(-b*x^2/a)^(1/2)*(c+d*(b*x^2+a)^(1/2))^(1/2)*Ellipti 
cE(1/2*(1-(b*x^2+a)^(1/2)/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(c+a^( 
1/2)*d))^(1/2))/b/d/x/((c+d*(b*x^2+a)^(1/2))/(c+a^(1/2)*d))^(1/2)+2/5*a^(1 
/2)*c*(-a*d^2+c^2)*(-b*x^2/a)^(1/2)*((c+d*(b*x^2+a)^(1/2))/(c+a^(1/2)*d))^ 
(1/2)*EllipticF(1/2*(1-(b*x^2+a)^(1/2)/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^( 
1/2)*d/(c+a^(1/2)*d))^(1/2))/b/d/x/(c+d*(b*x^2+a)^(1/2))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.16 (sec) , antiderivative size = 950, normalized size of antiderivative = 2.84 \[ \int \left (c+d \sqrt {a+b x^2}\right )^{3/2} \, dx=\frac {-2 b d^2 \sqrt {-c-\sqrt {a} d} \left (2 c^2-9 a d^2\right ) x^2+2 b d^2 \sqrt {-c-\sqrt {a} d} x^2 \left (c+d \sqrt {a+b x^2}\right ) \left (c+3 d \sqrt {a+b x^2}\right )+2 i \left (2 c^3+2 \sqrt {a} c^2 d-9 a c d^2-9 a^{3/2} d^3\right ) \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right )|\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )+2 i \sqrt {a} d \left (-2 c^2+7 \sqrt {a} c d+9 a d^2\right ) \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right ),\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )+5 c \left (2 b d^2 \sqrt {-c-\sqrt {a} d} x^2 \left (2 c+d \sqrt {a+b x^2}\right )-2 i c \left (c+\sqrt {a} d\right ) \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right )|\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )+2 i \sqrt {a} d \left (c+\sqrt {a} d\right ) \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right ),\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )\right )}{15 b d^2 \sqrt {-c-\sqrt {a} d} x \sqrt {c+d \sqrt {a+b x^2}}} \] Input:

Integrate[(c + d*Sqrt[a + b*x^2])^(3/2),x]
 

Output:

(-2*b*d^2*Sqrt[-c - Sqrt[a]*d]*(2*c^2 - 9*a*d^2)*x^2 + 2*b*d^2*Sqrt[-c - S 
qrt[a]*d]*x^2*(c + d*Sqrt[a + b*x^2])*(c + 3*d*Sqrt[a + b*x^2]) + (2*I)*(2 
*c^3 + 2*Sqrt[a]*c^2*d - 9*a*c*d^2 - 9*a^(3/2)*d^3)*Sqrt[(d*(-Sqrt[a] + Sq 
rt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*Sqrt[(d*(Sqrt[a] + Sqrt[a + b*x^2 
]))/(c + d*Sqrt[a + b*x^2])]*(c + d*Sqrt[a + b*x^2])^(3/2)*EllipticE[I*Arc 
Sinh[Sqrt[-c - Sqrt[a]*d]/Sqrt[c + d*Sqrt[a + b*x^2]]], (c - Sqrt[a]*d)/(c 
 + Sqrt[a]*d)] + (2*I)*Sqrt[a]*d*(-2*c^2 + 7*Sqrt[a]*c*d + 9*a*d^2)*Sqrt[( 
d*(-Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*Sqrt[(d*(Sqrt[a] 
+ Sqrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*(c + d*Sqrt[a + b*x^2])^(3/2) 
*EllipticF[I*ArcSinh[Sqrt[-c - Sqrt[a]*d]/Sqrt[c + d*Sqrt[a + b*x^2]]], (c 
 - Sqrt[a]*d)/(c + Sqrt[a]*d)] + 5*c*(2*b*d^2*Sqrt[-c - Sqrt[a]*d]*x^2*(2* 
c + d*Sqrt[a + b*x^2]) - (2*I)*c*(c + Sqrt[a]*d)*Sqrt[(d*(-Sqrt[a] + Sqrt[ 
a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*Sqrt[(d*(Sqrt[a] + Sqrt[a + b*x^2])) 
/(c + d*Sqrt[a + b*x^2])]*(c + d*Sqrt[a + b*x^2])^(3/2)*EllipticE[I*ArcSin 
h[Sqrt[-c - Sqrt[a]*d]/Sqrt[c + d*Sqrt[a + b*x^2]]], (c - Sqrt[a]*d)/(c + 
Sqrt[a]*d)] + (2*I)*Sqrt[a]*d*(c + Sqrt[a]*d)*Sqrt[(d*(-Sqrt[a] + Sqrt[a + 
 b*x^2]))/(c + d*Sqrt[a + b*x^2])]*Sqrt[(d*(Sqrt[a] + Sqrt[a + b*x^2]))/(c 
 + d*Sqrt[a + b*x^2])]*(c + d*Sqrt[a + b*x^2])^(3/2)*EllipticF[I*ArcSinh[S 
qrt[-c - Sqrt[a]*d]/Sqrt[c + d*Sqrt[a + b*x^2]]], (c - Sqrt[a]*d)/(c + Sqr 
t[a]*d)]))/(15*b*d^2*Sqrt[-c - Sqrt[a]*d]*x*Sqrt[c + d*Sqrt[a + b*x^2]]...
 

Rubi [F]

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.

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

\(\Big \downarrow \) 7299

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

Input:

Int[(c + d*Sqrt[a + b*x^2])^(3/2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int \left (c +d \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}d x\]

Input:

int((c+d*(b*x^2+a)^(1/2))^(3/2),x)
 

Output:

int((c+d*(b*x^2+a)^(1/2))^(3/2),x)
 

Fricas [F]

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

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

Output:

integral((sqrt(b*x^2 + a)*d + c)^(3/2), x)
 

Sympy [F]

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

integrate((c+d*(b*x**2+a)**(1/2))**(3/2),x)
 

Output:

Integral((c + d*sqrt(a + b*x**2))**(3/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int((c + d*(a + b*x^2)^(1/2))^(3/2),x)
 

Output:

int((c + d*(a + b*x^2)^(1/2))^(3/2), x)
 

Reduce [F]

\[ \int \left (c+d \sqrt {a+b x^2}\right )^{3/2} \, dx=\text {too large to display} \] Input:

int((c+d*(b*x^2+a)^(1/2))^(3/2),x)
 

Output:

(4*sqrt(a + b*x**2)*sqrt(sqrt(a + b*x**2)*d + c)*a*d**3*x - 2*sqrt(a + b*x 
**2)*sqrt(sqrt(a + b*x**2)*d + c)*c**2*d*x + 49*int(sqrt(sqrt(a + b*x**2)* 
d + c)/(7*a**3*d**4 + 14*a**2*b*d**4*x**2 - 11*a**2*c**2*d**2 + 7*a*b**2*d 
**4*x**4 - 15*a*b*c**2*d**2*x**2 + 4*a*c**4 - 4*b**2*c**2*d**2*x**4 + 4*b* 
c**4*x**2),x)*a**4*c*d**6 - 105*int(sqrt(sqrt(a + b*x**2)*d + c)/(7*a**3*d 
**4 + 14*a**2*b*d**4*x**2 - 11*a**2*c**2*d**2 + 7*a*b**2*d**4*x**4 - 15*a* 
b*c**2*d**2*x**2 + 4*a*c**4 - 4*b**2*c**2*d**2*x**4 + 4*b*c**4*x**2),x)*a* 
*3*c**3*d**4 + 72*int(sqrt(sqrt(a + b*x**2)*d + c)/(7*a**3*d**4 + 14*a**2* 
b*d**4*x**2 - 11*a**2*c**2*d**2 + 7*a*b**2*d**4*x**4 - 15*a*b*c**2*d**2*x* 
*2 + 4*a*c**4 - 4*b**2*c**2*d**2*x**4 + 4*b*c**4*x**2),x)*a**2*c**5*d**2 - 
 16*int(sqrt(sqrt(a + b*x**2)*d + c)/(7*a**3*d**4 + 14*a**2*b*d**4*x**2 - 
11*a**2*c**2*d**2 + 7*a*b**2*d**4*x**4 - 15*a*b*c**2*d**2*x**2 + 4*a*c**4 
- 4*b**2*c**2*d**2*x**4 + 4*b*c**4*x**2),x)*a*c**7 + 63*int((sqrt(sqrt(a + 
 b*x**2)*d + c)*x**4)/(7*a**3*d**4 + 14*a**2*b*d**4*x**2 - 11*a**2*c**2*d* 
*2 + 7*a*b**2*d**4*x**4 - 15*a*b*c**2*d**2*x**2 + 4*a*c**4 - 4*b**2*c**2*d 
**2*x**4 + 4*b*c**4*x**2),x)*a**2*b**2*c*d**6 - 71*int((sqrt(sqrt(a + b*x* 
*2)*d + c)*x**4)/(7*a**3*d**4 + 14*a**2*b*d**4*x**2 - 11*a**2*c**2*d**2 + 
7*a*b**2*d**4*x**4 - 15*a*b*c**2*d**2*x**2 + 4*a*c**4 - 4*b**2*c**2*d**2*x 
**4 + 4*b*c**4*x**2),x)*a*b**2*c**3*d**4 + 20*int((sqrt(sqrt(a + b*x**2)*d 
 + c)*x**4)/(7*a**3*d**4 + 14*a**2*b*d**4*x**2 - 11*a**2*c**2*d**2 + 7*...