3.32.26 \(\int \frac {(-d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{d+c x} \, dx\) [3126]

3.32.26.1 Optimal result

Integrand size = 38, antiderivative size = 747 \[ \int \frac {(-d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{d+c x} \, dx=\frac {2 (-6 a d+a c x) \sqrt {b^2+a^2 x^2}+2 \left (-b^2 c-6 a^2 d x+a^2 c x^2\right )}{3 a c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {4 a d^2 \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{c^{3/2} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}-\frac {4 b^2 \sqrt {c} d \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {b^2 c^2+a^2 d^2} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}-\frac {4 a^2 d^3 \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{c^{3/2} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {4 a d^2 \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{c^{3/2} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}+\frac {4 b^2 \sqrt {c} d \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {b^2 c^2+a^2 d^2} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}+\frac {4 a^2 d^3 \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{c^{3/2} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}} \]

output

1/3*(2*(a*c*x-6*a*d)*(a^2*x^2+b^2)^(1/2)+2*a^2*c*x^2-12*a^2*d*x-2*b^2*c)/a 
/c/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)+4*a*d^2*arctan(c^(1/2)*(a*x+(a^2*x^2+b^ 
2)^(1/2))^(1/2)/(a*d-(a^2*d^2+b^2*c^2)^(1/2))^(1/2))/c^(3/2)/(a*d-(a^2*d^2 
+b^2*c^2)^(1/2))^(1/2)-4*b^2*c^(1/2)*d*arctan(c^(1/2)*(a*x+(a^2*x^2+b^2)^( 
1/2))^(1/2)/(a*d-(a^2*d^2+b^2*c^2)^(1/2))^(1/2))/(a^2*d^2+b^2*c^2)^(1/2)/( 
a*d-(a^2*d^2+b^2*c^2)^(1/2))^(1/2)-4*a^2*d^3*arctan(c^(1/2)*(a*x+(a^2*x^2+ 
b^2)^(1/2))^(1/2)/(a*d-(a^2*d^2+b^2*c^2)^(1/2))^(1/2))/c^(3/2)/(a^2*d^2+b^ 
2*c^2)^(1/2)/(a*d-(a^2*d^2+b^2*c^2)^(1/2))^(1/2)+4*a*d^2*arctan(c^(1/2)*(a 
*x+(a^2*x^2+b^2)^(1/2))^(1/2)/(a*d+(a^2*d^2+b^2*c^2)^(1/2))^(1/2))/c^(3/2) 
/(a*d+(a^2*d^2+b^2*c^2)^(1/2))^(1/2)+4*b^2*c^(1/2)*d*arctan(c^(1/2)*(a*x+( 
a^2*x^2+b^2)^(1/2))^(1/2)/(a*d+(a^2*d^2+b^2*c^2)^(1/2))^(1/2))/(a^2*d^2+b^ 
2*c^2)^(1/2)/(a*d+(a^2*d^2+b^2*c^2)^(1/2))^(1/2)+4*a^2*d^3*arctan(c^(1/2)* 
(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)/(a*d+(a^2*d^2+b^2*c^2)^(1/2))^(1/2))/c^(3/ 
2)/(a^2*d^2+b^2*c^2)^(1/2)/(a*d+(a^2*d^2+b^2*c^2)^(1/2))^(1/2)
 

3.32.26.2 Mathematica [A] (verified)

Time = 1.80 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.49 \[ \int \frac {(-d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{d+c x} \, dx=\frac {2 \left (\frac {\sqrt {c} \sqrt {b^2 c^2+a^2 d^2} \left (-b^2 c+a (-6 d+c x) \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {6 d \left (-b^2 c^2+a d \left (-a d+\sqrt {b^2 c^2+a^2 d^2}\right )\right ) \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {6 d \left (b^2 c^2+a d \left (a d+\sqrt {b^2 c^2+a^2 d^2}\right )\right ) \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{3 c^{3/2} \sqrt {b^2 c^2+a^2 d^2}} \]

input

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

output

(2*((Sqrt[c]*Sqrt[b^2*c^2 + a^2*d^2]*(-(b^2*c) + a*(-6*d + c*x)*(a*x + Sqr 
t[b^2 + a^2*x^2])))/(a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]) + (6*d*(-(b^2*c^2) 
 + a*d*(-(a*d) + Sqrt[b^2*c^2 + a^2*d^2]))*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt 
[b^2 + a^2*x^2]])/Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]])/Sqrt[a*d - Sqrt[b^ 
2*c^2 + a^2*d^2]] + (6*d*(b^2*c^2 + a*d*(a*d + Sqrt[b^2*c^2 + a^2*d^2]))*A 
rcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d + Sqrt[b^2*c^2 + 
a^2*d^2]]])/Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]]))/(3*c^(3/2)*Sqrt[b^2*c^2 
+ a^2*d^2])
 

3.32.26.3 Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {7293, 2009}

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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

-(b^2/(a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) - (4*d*Sqrt[a*x + Sqrt[b^2 + a^ 
2*x^2]])/c + (a*x + Sqrt[b^2 + a^2*x^2])^(3/2)/(3*a) + (4*d*Sqrt[a*d - Sqr 
t[b^2*c^2 + a^2*d^2]]*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqr 
t[a*d - Sqrt[b^2*c^2 + a^2*d^2]]])/c^(3/2) + (4*d*Sqrt[a*d + Sqrt[b^2*c^2 
+ a^2*d^2]]*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d + Sq 
rt[b^2*c^2 + a^2*d^2]]])/c^(3/2)
 

3.32.26.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.32.26.4 Maple [F]

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

input

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

output

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

3.32.26.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 511, normalized size of antiderivative = 0.68 \[ \int \frac {(-d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{d+c x} \, dx=\frac {2 \, {\left (3 \, a c \sqrt {-\frac {a d^{3} + c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}} \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} d + 4 \, c \sqrt {-\frac {a d^{3} + c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}}\right ) - 3 \, a c \sqrt {-\frac {a d^{3} + c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}} \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} d - 4 \, c \sqrt {-\frac {a d^{3} + c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}}\right ) + 3 \, a c \sqrt {-\frac {a d^{3} - c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}} \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} d + 4 \, c \sqrt {-\frac {a d^{3} - c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}}\right ) - 3 \, a c \sqrt {-\frac {a d^{3} - c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}} \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} d - 4 \, c \sqrt {-\frac {a d^{3} - c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}}\right ) + {\left (2 \, a c x - 6 \, a d - \sqrt {a^{2} x^{2} + b^{2}} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right )}}{3 \, a c} \]

input

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

output

2/3*(3*a*c*sqrt(-(a*d^3 + c^3*sqrt((b^2*c^2*d^4 + a^2*d^6)/c^6))/c^3)*log( 
4*sqrt(a*x + sqrt(a^2*x^2 + b^2))*d + 4*c*sqrt(-(a*d^3 + c^3*sqrt((b^2*c^2 
*d^4 + a^2*d^6)/c^6))/c^3)) - 3*a*c*sqrt(-(a*d^3 + c^3*sqrt((b^2*c^2*d^4 + 
 a^2*d^6)/c^6))/c^3)*log(4*sqrt(a*x + sqrt(a^2*x^2 + b^2))*d - 4*c*sqrt(-( 
a*d^3 + c^3*sqrt((b^2*c^2*d^4 + a^2*d^6)/c^6))/c^3)) + 3*a*c*sqrt(-(a*d^3 
- c^3*sqrt((b^2*c^2*d^4 + a^2*d^6)/c^6))/c^3)*log(4*sqrt(a*x + sqrt(a^2*x^ 
2 + b^2))*d + 4*c*sqrt(-(a*d^3 - c^3*sqrt((b^2*c^2*d^4 + a^2*d^6)/c^6))/c^ 
3)) - 3*a*c*sqrt(-(a*d^3 - c^3*sqrt((b^2*c^2*d^4 + a^2*d^6)/c^6))/c^3)*log 
(4*sqrt(a*x + sqrt(a^2*x^2 + b^2))*d - 4*c*sqrt(-(a*d^3 - c^3*sqrt((b^2*c^ 
2*d^4 + a^2*d^6)/c^6))/c^3)) + (2*a*c*x - 6*a*d - sqrt(a^2*x^2 + b^2)*c)*s 
qrt(a*x + sqrt(a^2*x^2 + b^2)))/(a*c)
 

3.32.26.6 Sympy [F]

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

input

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

output

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

3.32.26.7 Maxima [F]

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

input

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

output

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

3.32.26.8 Giac [F]

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

input

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

output

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

3.32.26.9 Mupad [F(-1)]

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

input

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

output

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