\(\int \frac {(a+b x^2)^{3/2}}{x^6 (c+d x)} \, dx\) [1084]

Optimal result

Integrand size = 22, antiderivative size = 284 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^6 (c+d x)} \, dx=-\frac {a \sqrt {a+b x^2}}{5 c x^5}+\frac {a d \sqrt {a+b x^2}}{4 c^2 x^4}-\frac {\left (6 b c^2+5 a d^2\right ) \sqrt {a+b x^2}}{15 c^3 x^3}+\frac {d \left (5 b c^2+4 a d^2\right ) \sqrt {a+b x^2}}{8 c^4 x^2}-\frac {\left (3 b^2 c^4+20 a b c^2 d^2+15 a^2 d^4\right ) \sqrt {a+b x^2}}{15 a c^5 x}-\frac {d^2 \left (b c^2+a d^2\right )^{3/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^6}+\frac {d \left (3 b^2 c^4+12 a b c^2 d^2+8 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a} c^6} \] Output:

-1/5*a*(b*x^2+a)^(1/2)/c/x^5+1/4*a*d*(b*x^2+a)^(1/2)/c^2/x^4-1/15*(5*a*d^2 
+6*b*c^2)*(b*x^2+a)^(1/2)/c^3/x^3+1/8*d*(4*a*d^2+5*b*c^2)*(b*x^2+a)^(1/2)/ 
c^4/x^2-1/15*(15*a^2*d^4+20*a*b*c^2*d^2+3*b^2*c^4)*(b*x^2+a)^(1/2)/a/c^5/x 
-d^2*(a*d^2+b*c^2)^(3/2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a 
)^(1/2))/c^6+1/8*d*(8*a^2*d^4+12*a*b*c^2*d^2+3*b^2*c^4)*arctanh((b*x^2+a)^ 
(1/2)/a^(1/2))/a^(1/2)/c^6
 

Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^6 (c+d x)} \, dx=-\frac {\frac {c \sqrt {a+b x^2} \left (24 b^2 c^4 x^4+a b c^2 x^2 \left (48 c^2-75 c d x+160 d^2 x^2\right )+2 a^2 \left (12 c^4-15 c^3 d x+20 c^2 d^2 x^2-30 c d^3 x^3+60 d^4 x^4\right )\right )}{a x^5}+240 d^2 \left (-b c^2-a d^2\right )^{3/2} \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )+\frac {30 d \left (3 b^2 c^4+12 a b c^2 d^2+8 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}}{120 c^6} \] Input:

Integrate[(a + b*x^2)^(3/2)/(x^6*(c + d*x)),x]
 

Output:

-1/120*((c*Sqrt[a + b*x^2]*(24*b^2*c^4*x^4 + a*b*c^2*x^2*(48*c^2 - 75*c*d* 
x + 160*d^2*x^2) + 2*a^2*(12*c^4 - 15*c^3*d*x + 20*c^2*d^2*x^2 - 30*c*d^3* 
x^3 + 60*d^4*x^4)))/(a*x^5) + 240*d^2*(-(b*c^2) - a*d^2)^(3/2)*ArcTan[(Sqr 
t[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]] + (30*d*(3*b^2 
*c^4 + 12*a*b*c^2*d^2 + 8*a^2*d^4)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/S 
qrt[a]])/Sqrt[a])/c^6
 

Rubi [A] (verified)

Time = 1.30 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{x^6 (c+d x)} \, dx\)

\(\Big \downarrow \) 617

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^{3/2} d^5 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^6}+\frac {b^{3/2} d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{c^3}+\frac {3 b^2 d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a} c^2}+\frac {3 a \sqrt {b} d^4 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 c^5}+\frac {3 \sqrt {a} b d^3 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 c^4}-\frac {d^2 \left (a d^2+b c^2\right )^{3/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^6}-\frac {\sqrt {b} d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a d^2+2 b c^2\right )}{2 c^5}-\frac {a d^5 \sqrt {a+b x^2}}{c^6}-\frac {d^4 \left (a+b x^2\right )^{3/2}}{c^5 x}+\frac {3 b d^4 x \sqrt {a+b x^2}}{2 c^5}+\frac {d^3 \left (a+b x^2\right )^{3/2}}{2 c^4 x^2}-\frac {3 b d^3 \sqrt {a+b x^2}}{2 c^4}-\frac {b d^2 \sqrt {a+b x^2}}{c^3 x}-\frac {d^2 \left (a+b x^2\right )^{3/2}}{3 c^3 x^3}+\frac {3 b d \sqrt {a+b x^2}}{8 c^2 x^2}+\frac {d \left (a+b x^2\right )^{3/2}}{4 c^2 x^4}+\frac {d^3 \sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )-b c d x\right )}{2 c^6}-\frac {\left (a+b x^2\right )^{5/2}}{5 a c x^5}\)

Input:

Int[(a + b*x^2)^(3/2)/(x^6*(c + d*x)),x]
 

Output:

(-3*b*d^3*Sqrt[a + b*x^2])/(2*c^4) - (a*d^5*Sqrt[a + b*x^2])/c^6 + (3*b*d* 
Sqrt[a + b*x^2])/(8*c^2*x^2) - (b*d^2*Sqrt[a + b*x^2])/(c^3*x) + (3*b*d^4* 
x*Sqrt[a + b*x^2])/(2*c^5) + (d^3*(2*(b*c^2 + a*d^2) - b*c*d*x)*Sqrt[a + b 
*x^2])/(2*c^6) + (d*(a + b*x^2)^(3/2))/(4*c^2*x^4) - (d^2*(a + b*x^2)^(3/2 
))/(3*c^3*x^3) + (d^3*(a + b*x^2)^(3/2))/(2*c^4*x^2) - (d^4*(a + b*x^2)^(3 
/2))/(c^5*x) - (a + b*x^2)^(5/2)/(5*a*c*x^5) + (b^(3/2)*d^2*ArcTanh[(Sqrt[ 
b]*x)/Sqrt[a + b*x^2]])/c^3 + (3*a*Sqrt[b]*d^4*ArcTanh[(Sqrt[b]*x)/Sqrt[a 
+ b*x^2]])/(2*c^5) - (Sqrt[b]*d^2*(2*b*c^2 + 3*a*d^2)*ArcTanh[(Sqrt[b]*x)/ 
Sqrt[a + b*x^2]])/(2*c^5) - (d^2*(b*c^2 + a*d^2)^(3/2)*ArcTanh[(a*d - b*c* 
x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/c^6 + (3*b^2*d*ArcTanh[Sqrt[a + 
 b*x^2]/Sqrt[a]])/(8*Sqrt[a]*c^2) + (3*Sqrt[a]*b*d^3*ArcTanh[Sqrt[a + b*x^ 
2]/Sqrt[a]])/(2*c^4) + (a^(3/2)*d^5*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/c^6
 

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 2009

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

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.19

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (120 a^{2} d^{4} x^{4}+160 x^{4} a b \,c^{2} d^{2}+24 x^{4} b^{2} c^{4}-60 a^{2} c \,d^{3} x^{3}-75 a b \,c^{3} d \,x^{3}+40 a^{2} c^{2} d^{2} x^{2}+48 a b \,c^{4} x^{2}-30 a^{2} d \,c^{3} x +24 a^{2} c^{4}\right )}{120 a \,c^{5} x^{5}}+\frac {d \left (\frac {\left (8 a^{2} d^{4}+12 b \,c^{2} d^{2} a +3 b^{2} c^{4}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}-\frac {8 \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{8 c^{5}}\)	\(338\)
default	\(\text {Expression too large to display}\)	\(958\)

Input:

int((b*x^2+a)^(3/2)/x^6/(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

-1/120*(b*x^2+a)^(1/2)*(120*a^2*d^4*x^4+160*a*b*c^2*d^2*x^4+24*b^2*c^4*x^4 
-60*a^2*c*d^3*x^3-75*a*b*c^3*d*x^3+40*a^2*c^2*d^2*x^2+48*a*b*c^4*x^2-30*a^ 
2*c^3*d*x+24*a^2*c^4)/a/c^5/x^5+1/8*d/c^5*((8*a^2*d^4+12*a*b*c^2*d^2+3*b^2 
*c^4)/c/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-8*(a^2*d^4+2*a*b*c^2 
*d^2+b^2*c^4)/c/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d* 
(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b* 
c^2)/d^2)^(1/2))/(x+c/d)))
 

Fricas [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 1215, normalized size of antiderivative = 4.28 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^6 (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)/x^6/(d*x+c),x, algorithm="fricas")
 

Output:

[1/240*(120*(a*b*c^2*d^2 + a^2*d^4)*sqrt(b*c^2 + a*d^2)*x^5*log((2*a*b*c*d 
*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 - 2*sqrt(b*c^2 + a*d^ 
2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) + 15*(3*b^2*c 
^4*d + 12*a*b*c^2*d^3 + 8*a^2*d^5)*sqrt(a)*x^5*log(-(b*x^2 + 2*sqrt(b*x^2 
+ a)*sqrt(a) + 2*a)/x^2) + 2*(30*a^2*c^4*d*x - 24*a^2*c^5 - 8*(3*b^2*c^5 + 
 20*a*b*c^3*d^2 + 15*a^2*c*d^4)*x^4 + 15*(5*a*b*c^4*d + 4*a^2*c^2*d^3)*x^3 
 - 8*(6*a*b*c^5 + 5*a^2*c^3*d^2)*x^2)*sqrt(b*x^2 + a))/(a*c^6*x^5), -1/240 
*(240*(a*b*c^2*d^2 + a^2*d^4)*sqrt(-b*c^2 - a*d^2)*x^5*arctan(sqrt(-b*c^2 
- a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b 
*d^2)*x^2)) - 15*(3*b^2*c^4*d + 12*a*b*c^2*d^3 + 8*a^2*d^5)*sqrt(a)*x^5*lo 
g(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(30*a^2*c^4*d*x - 24 
*a^2*c^5 - 8*(3*b^2*c^5 + 20*a*b*c^3*d^2 + 15*a^2*c*d^4)*x^4 + 15*(5*a*b*c 
^4*d + 4*a^2*c^2*d^3)*x^3 - 8*(6*a*b*c^5 + 5*a^2*c^3*d^2)*x^2)*sqrt(b*x^2 
+ a))/(a*c^6*x^5), -1/120*(15*(3*b^2*c^4*d + 12*a*b*c^2*d^3 + 8*a^2*d^5)*s 
qrt(-a)*x^5*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - 60*(a*b*c^2*d^2 + a^2*d^4 
)*sqrt(b*c^2 + a*d^2)*x^5*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2* 
c^2 + a*b*d^2)*x^2 - 2*sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/ 
(d^2*x^2 + 2*c*d*x + c^2)) - (30*a^2*c^4*d*x - 24*a^2*c^5 - 8*(3*b^2*c^5 + 
 20*a*b*c^3*d^2 + 15*a^2*c*d^4)*x^4 + 15*(5*a*b*c^4*d + 4*a^2*c^2*d^3)*x^3 
 - 8*(6*a*b*c^5 + 5*a^2*c^3*d^2)*x^2)*sqrt(b*x^2 + a))/(a*c^6*x^5), -1/...
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^6 (c+d x)} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{x^{6} \left (c + d x\right )}\, dx \] Input:

integrate((b*x**2+a)**(3/2)/x**6/(d*x+c),x)
 

Output:

Integral((a + b*x**2)**(3/2)/(x**6*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^6 (c+d x)} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )} x^{6}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/x^6/(d*x+c),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(3/2)/((d*x + c)*x^6), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 758 vs. \(2 (251) = 502\).

Time = 0.15 (sec) , antiderivative size = 758, normalized size of antiderivative = 2.67 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^6 (c+d x)} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)/x^6/(d*x+c),x, algorithm="giac")
 

Output:

2*(b^2*c^4*d^2 + 2*a*b*c^2*d^4 + a^2*d^6)*arctan(-((sqrt(b)*x - sqrt(b*x^2 
 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/(sqrt(-b*c^2 - a*d^2)*c^6) - 1 
/4*(3*b^2*c^4*d + 12*a*b*c^2*d^3 + 8*a^2*d^5)*arctan(-(sqrt(b)*x - sqrt(b* 
x^2 + a))/sqrt(-a))/(sqrt(-a)*c^6) - 1/60*(75*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^9*b^2*c^3*d + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^9*a*b*c*d^3 - 120*(sqrt(b 
)*x - sqrt(b*x^2 + a))^8*b^(5/2)*c^4 - 240*(sqrt(b)*x - sqrt(b*x^2 + a))^8 
*a*b^(3/2)*c^2*d^2 - 120*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*sqrt(b)*d^4 - 
 30*(sqrt(b)*x - sqrt(b*x^2 + a))^7*a*b^2*c^3*d - 120*(sqrt(b)*x - sqrt(b* 
x^2 + a))^7*a^2*b*c*d^3 + 720*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(3/2)* 
c^2*d^2 + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*sqrt(b)*d^4 - 240*(sqrt( 
b)*x - sqrt(b*x^2 + a))^4*a^2*b^(5/2)*c^4 - 880*(sqrt(b)*x - sqrt(b*x^2 + 
a))^4*a^3*b^(3/2)*c^2*d^2 - 720*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*sqrt(b 
)*d^4 + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a^3*b^2*c^3*d + 120*(sqrt(b)*x 
- sqrt(b*x^2 + a))^3*a^4*b*c*d^3 + 560*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4 
*b^(3/2)*c^2*d^2 + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*sqrt(b)*d^4 - 7 
5*(sqrt(b)*x - sqrt(b*x^2 + a))*a^4*b^2*c^3*d - 60*(sqrt(b)*x - sqrt(b*x^2 
 + a))*a^5*b*c*d^3 - 24*a^4*b^(5/2)*c^4 - 160*a^5*b^(3/2)*c^2*d^2 - 120*a^ 
6*sqrt(b)*d^4)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5*c^5)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^6 (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{x^6\,\left (c+d\,x\right )} \,d x \] Input:

int((a + b*x^2)^(3/2)/(x^6*(c + d*x)),x)
 

Output:

int((a + b*x^2)^(3/2)/(x^6*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^6 (c+d x)} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{x^{6} \left (d x +c \right )}d x \] Input:

int((b*x^2+a)^(3/2)/x^6/(d*x+c),x)
 

Output:

int((b*x^2+a)^(3/2)/x^6/(d*x+c),x)