3.30.88 \(\int \frac {\sqrt {-b+a^2 x^2} (d+c x^4) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx\) [2988]

3.30.88.1 Optimal result

Integrand size = 49, antiderivative size = 390 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\frac {63 a^3 b^3 d-72 a b^4 c x^2+126 a^5 b^2 d x^2+210 a^3 b^3 c x^4-2016 a^7 b d x^4+126 a^5 b^2 c x^6+2016 a^9 d x^6-504 a^7 b c x^8+224 a^9 c x^{10}+\sqrt {-b+a^2 x^2} \left (-16 b^4 c x-126 a^4 b^2 d x+154 a^2 b^3 c x^3-1008 a^6 b d x^3-42 a^4 b^2 c x^5+2016 a^8 d x^5-392 a^6 b c x^7+224 a^8 c x^9\right )}{63 a^3 x \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}+\frac {a \sqrt [4]{b} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{-\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-\frac {a \sqrt [4]{b} d \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {a x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt {-b+a^2 x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{\sqrt {2}} \]

output

1/63*(63*a^3*b^3*d-72*a*b^4*c*x^2+126*a^5*b^2*d*x^2+210*a^3*b^3*c*x^4-2016 
*a^7*b*d*x^4+126*a^5*b^2*c*x^6+2016*a^9*d*x^6-504*a^7*b*c*x^8+224*a^9*c*x^ 
10+(a^2*x^2-b)^(1/2)*(224*a^8*c*x^9-392*a^6*b*c*x^7+2016*a^8*d*x^5-42*a^4* 
b^2*c*x^5-1008*a^6*b*d*x^3+154*a^2*b^3*c*x^3-126*a^4*b^2*d*x-16*b^4*c*x))/ 
a^3/x/(a*x+(a^2*x^2-b)^(1/2))^(9/2)+1/2*a*b^(1/4)*d*arctan(2^(1/2)*b^(1/4) 
*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(-b^(1/2)+a*x+(a^2*x^2-b)^(1/2)))*2^(1/2)-1 
/2*a*b^(1/4)*d*arctanh((1/2*b^(1/4)*2^(1/2)+1/2*a*x*2^(1/2)/b^(1/4)+1/2*(a 
^2*x^2-b)^(1/2)*2^(1/2)/b^(1/4))/(a*x+(a^2*x^2-b)^(1/2))^(1/2))*2^(1/2)
 

3.30.88.2 Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\frac {63 a^3 b^3 d-72 a b^4 c x^2+126 a^5 b^2 d x^2+210 a^3 b^3 c x^4-2016 a^7 b d x^4+126 a^5 b^2 c x^6+2016 a^9 d x^6-504 a^7 b c x^8+224 a^9 c x^{10}-2 x \sqrt {-b+a^2 x^2} \left (8 b^4 c-77 a^2 b^3 c x^2+21 a^4 b^2 \left (3 d+c x^4\right )-112 a^8 x^4 \left (9 d+c x^4\right )+28 a^6 b x^2 \left (18 d+7 c x^4\right )\right )}{63 a^3 x \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}+\frac {a \sqrt [4]{b} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{-\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-\frac {a \sqrt [4]{b} d \text {arctanh}\left (\frac {\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{\sqrt {2}} \]

input

Integrate[(Sqrt[-b + a^2*x^2]*(d + c*x^4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/ 
x^2,x]
 

output

(63*a^3*b^3*d - 72*a*b^4*c*x^2 + 126*a^5*b^2*d*x^2 + 210*a^3*b^3*c*x^4 - 2 
016*a^7*b*d*x^4 + 126*a^5*b^2*c*x^6 + 2016*a^9*d*x^6 - 504*a^7*b*c*x^8 + 2 
24*a^9*c*x^10 - 2*x*Sqrt[-b + a^2*x^2]*(8*b^4*c - 77*a^2*b^3*c*x^2 + 21*a^ 
4*b^2*(3*d + c*x^4) - 112*a^8*x^4*(9*d + c*x^4) + 28*a^6*b*x^2*(18*d + 7*c 
*x^4)))/(63*a^3*x*(a*x + Sqrt[-b + a^2*x^2])^(9/2)) + (a*b^(1/4)*d*ArcTan[ 
(Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/(-Sqrt[b] + a*x + Sqrt[-b 
 + a^2*x^2])])/Sqrt[2] - (a*b^(1/4)*d*ArcTanh[(Sqrt[b] + a*x + Sqrt[-b + a 
^2*x^2])/(Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/Sqrt[2]
 

3.30.88.3 Rubi [A] (verified)

Time = 1.89 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a \sqrt [4]{b} d \arctan \left (1-\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}\right )}{\sqrt {2}}-\frac {a \sqrt [4]{b} d \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2}}+2 a d \sqrt {\sqrt {a^2 x^2-b}+a x}+\frac {2 a b d \sqrt {\sqrt {a^2 x^2-b}+a x}}{\left (\sqrt {a^2 x^2-b}+a x\right )^2+b}+\frac {a \sqrt [4]{b} d \log \left (\sqrt {a^2 x^2-b}-\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}+a x+\sqrt {b}\right )}{2 \sqrt {2}}-\frac {a \sqrt [4]{b} d \log \left (\sqrt {a^2 x^2-b}+\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}+a x+\sqrt {b}\right )}{2 \sqrt {2}}-\frac {b^4 c}{56 a^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{7/2}}-\frac {b^2 c \sqrt {\sqrt {a^2 x^2-b}+a x}}{4 a^3}+\frac {c \left (\sqrt {a^2 x^2-b}+a x\right )^{9/2}}{72 a^3}\)

input

Int[(Sqrt[-b + a^2*x^2]*(d + c*x^4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/x^2,x]
 

output

-1/56*(b^4*c)/(a^3*(a*x + Sqrt[-b + a^2*x^2])^(7/2)) - (b^2*c*Sqrt[a*x + S 
qrt[-b + a^2*x^2]])/(4*a^3) + 2*a*d*Sqrt[a*x + Sqrt[-b + a^2*x^2]] + (c*(a 
*x + Sqrt[-b + a^2*x^2])^(9/2))/(72*a^3) + (2*a*b*d*Sqrt[a*x + Sqrt[-b + a 
^2*x^2]])/(b + (a*x + Sqrt[-b + a^2*x^2])^2) + (a*b^(1/4)*d*ArcTan[1 - (Sq 
rt[2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/4)])/Sqrt[2] - (a*b^(1/4)*d*Arc 
Tan[1 + (Sqrt[2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/4)])/Sqrt[2] + (a*b^ 
(1/4)*d*Log[Sqrt[b] + a*x + Sqrt[-b + a^2*x^2] - Sqrt[2]*b^(1/4)*Sqrt[a*x 
+ Sqrt[-b + a^2*x^2]]])/(2*Sqrt[2]) - (a*b^(1/4)*d*Log[Sqrt[b] + a*x + Sqr 
t[-b + a^2*x^2] + Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(2*Sqrt 
[2])
 

3.30.88.3.1 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] 
]
 

3.30.88.4 Maple [F]

\[\int \frac {\sqrt {a^{2} x^{2}-b}\, \left (c \,x^{4}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}-b}}}{x^{2}}d x\]

input

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

output

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

3.30.88.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=-\frac {63 \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}} a^{3} x \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a d + \left (-a^{4} b d^{4}\right )^{\frac {1}{4}}\right ) + 63 i \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}} a^{3} x \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a d + i \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}}\right ) - 63 i \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}} a^{3} x \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a d - i \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}}\right ) - 63 \, \left (-a^{4} b d^{4}\right )^{\frac {1}{4}} a^{3} x \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a d - \left (-a^{4} b d^{4}\right )^{\frac {1}{4}}\right ) + 2 \, {\left (2 \, a^{4} c x^{5} - 2 \, a^{2} b c x^{3} - {\left (189 \, a^{4} d - 16 \, b^{2} c\right )} x - {\left (16 \, a^{3} c x^{4} - 8 \, a b c x^{2} - 63 \, a^{3} d\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{126 \, a^{3} x} \]

input

integrate((a^2*x^2-b)^(1/2)*(c*x^4+d)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/x^2,x, 
 algorithm="fricas")
 

output

-1/126*(63*(-a^4*b*d^4)^(1/4)*a^3*x*log(sqrt(a*x + sqrt(a^2*x^2 - b))*a*d 
+ (-a^4*b*d^4)^(1/4)) + 63*I*(-a^4*b*d^4)^(1/4)*a^3*x*log(sqrt(a*x + sqrt( 
a^2*x^2 - b))*a*d + I*(-a^4*b*d^4)^(1/4)) - 63*I*(-a^4*b*d^4)^(1/4)*a^3*x* 
log(sqrt(a*x + sqrt(a^2*x^2 - b))*a*d - I*(-a^4*b*d^4)^(1/4)) - 63*(-a^4*b 
*d^4)^(1/4)*a^3*x*log(sqrt(a*x + sqrt(a^2*x^2 - b))*a*d - (-a^4*b*d^4)^(1/ 
4)) + 2*(2*a^4*c*x^5 - 2*a^2*b*c*x^3 - (189*a^4*d - 16*b^2*c)*x - (16*a^3* 
c*x^4 - 8*a*b*c*x^2 - 63*a^3*d)*sqrt(a^2*x^2 - b))*sqrt(a*x + sqrt(a^2*x^2 
 - b)))/(a^3*x)
 

3.30.88.6 Sympy [F]

\[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\int \frac {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b} \left (c x^{4} + d\right )}{x^{2}}\, dx \]

input

integrate((a**2*x**2-b)**(1/2)*(c*x**4+d)*(a*x+(a**2*x**2-b)**(1/2))**(1/2 
)/x**2,x)
 

output

Integral(sqrt(a*x + sqrt(a**2*x**2 - b))*sqrt(a**2*x**2 - b)*(c*x**4 + d)/ 
x**2, x)
 

3.30.88.7 Maxima [F]

\[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\int { \frac {{\left (c x^{4} + d\right )} \sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{x^{2}} \,d x } \]

input

integrate((a^2*x^2-b)^(1/2)*(c*x^4+d)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/x^2,x, 
 algorithm="maxima")
 

output

integrate((c*x^4 + d)*sqrt(a^2*x^2 - b)*sqrt(a*x + sqrt(a^2*x^2 - b))/x^2, 
 x)
 

3.30.88.8 Giac [F]

\[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\int { \frac {{\left (c x^{4} + d\right )} \sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{x^{2}} \,d x } \]

input

integrate((a^2*x^2-b)^(1/2)*(c*x^4+d)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/x^2,x, 
 algorithm="giac")
 

output

integrate((c*x^4 + d)*sqrt(a^2*x^2 - b)*sqrt(a*x + sqrt(a^2*x^2 - b))/x^2, 
 x)
 

3.30.88.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx=\int \frac {\sqrt {a\,x+\sqrt {a^2\,x^2-b}}\,\left (c\,x^4+d\right )\,\sqrt {a^2\,x^2-b}}{x^2} \,d x \]

input

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

output

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