\(\int \frac {(a+b x^2)^{7/2}}{x^6 (c+d x)} \, dx\) [1158]

Optimal result

Integrand size = 22, antiderivative size = 340 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x^6 (c+d x)} \, dx=\frac {b^3 \sqrt {a+b x^2}}{d}-\frac {a^3 \sqrt {a+b x^2}}{5 c x^5}+\frac {a^3 d \sqrt {a+b x^2}}{4 c^2 x^4}-\frac {a^2 \left (16 b c^2+5 a d^2\right ) \sqrt {a+b x^2}}{15 c^3 x^3}+\frac {a^2 d \left (13 b c^2+4 a d^2\right ) \sqrt {a+b x^2}}{8 c^4 x^2}-\frac {a \left (58 b^2 c^4+50 a b c^2 d^2+15 a^2 d^4\right ) \sqrt {a+b x^2}}{15 c^5 x}-\frac {b^{7/2} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^2}-\frac {\left (b c^2+a d^2\right )^{7/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^6 d^2}+\frac {a^{3/2} d \left (35 b^2 c^4+28 a b c^2 d^2+8 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 c^6} \] Output:

b^3*(b*x^2+a)^(1/2)/d-1/5*a^3*(b*x^2+a)^(1/2)/c/x^5+1/4*a^3*d*(b*x^2+a)^(1 
/2)/c^2/x^4-1/15*a^2*(5*a*d^2+16*b*c^2)*(b*x^2+a)^(1/2)/c^3/x^3+1/8*a^2*d* 
(4*a*d^2+13*b*c^2)*(b*x^2+a)^(1/2)/c^4/x^2-1/15*a*(15*a^2*d^4+50*a*b*c^2*d 
^2+58*b^2*c^4)*(b*x^2+a)^(1/2)/c^5/x-b^(7/2)*c*arctanh(b^(1/2)*x/(b*x^2+a) 
^(1/2))/d^2-(a*d^2+b*c^2)^(7/2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/( 
b*x^2+a)^(1/2))/c^6/d^2+1/8*a^(3/2)*d*(8*a^2*d^4+28*a*b*c^2*d^2+35*b^2*c^4 
)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/c^6
 

Mathematica [A] (verified)

Time = 2.34 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x^6 (c+d x)} \, dx=-\frac {\frac {c d \sqrt {a+b x^2} \left (464 a b^2 c^4 d x^4-120 b^3 c^5 x^5+a^2 b c^2 d x^2 \left (128 c^2-195 c d x+400 d^2 x^2\right )+2 a^3 d \left (12 c^4-15 c^3 d x+20 c^2 d^2 x^2-30 c d^3 x^3+60 d^4 x^4\right )\right )}{x^5}+240 \left (-b c^2-a d^2\right )^{7/2} \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )+30 a^{3/2} d^3 \left (35 b^2 c^4+28 a b c^2 d^2+8 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )-120 b^{7/2} c^7 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{120 c^6 d^2} \] Input:

Integrate[(a + b*x^2)^(7/2)/(x^6*(c + d*x)),x]
 

Output:

-1/120*((c*d*Sqrt[a + b*x^2]*(464*a*b^2*c^4*d*x^4 - 120*b^3*c^5*x^5 + a^2* 
b*c^2*d*x^2*(128*c^2 - 195*c*d*x + 400*d^2*x^2) + 2*a^3*d*(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)))/x^5 + 240*(-(b*c^2) - 
 a*d^2)^(7/2)*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) 
 - a*d^2]] + 30*a^(3/2)*d^3*(35*b^2*c^4 + 28*a*b*c^2*d^2 + 8*a^2*d^4)*ArcT 
anh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]] - 120*b^(7/2)*c^7*Log[-(Sqrt[b] 
*x) + Sqrt[a + b*x^2]])/(c^6*d^2)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1041\) vs. \(2(340)=680\).

Time = 2.21 (sec) , antiderivative size = 1041, normalized size of antiderivative = 3.06, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{7/2}}{x^6 (c+d x)} \, dx\)

\(\Big \downarrow \) 617

\(\displaystyle \int \left (\frac {d^6 \left (a+b x^2\right )^{7/2}}{c^6 (c+d x)}-\frac {d^5 \left (a+b x^2\right )^{7/2}}{c^6 x}+\frac {d^4 \left (a+b x^2\right )^{7/2}}{c^5 x^2}-\frac {d^3 \left (a+b x^2\right )^{7/2}}{c^4 x^3}+\frac {d^2 \left (a+b x^2\right )^{7/2}}{c^3 x^4}-\frac {d \left (a+b x^2\right )^{7/2}}{c^2 x^5}+\frac {\left (a+b x^2\right )^{7/2}}{c x^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a \left (b x^2+a\right )^{5/2} d^5}{5 c^6}-\frac {a^2 \left (b x^2+a\right )^{3/2} d^5}{3 c^6}+\frac {a^{7/2} \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d^5}{c^6}-\frac {a^3 \sqrt {b x^2+a} d^5}{c^6}-\frac {\left (b x^2+a\right )^{7/2} d^4}{c^5 x}+\frac {7 b x \left (b x^2+a\right )^{5/2} d^4}{6 c^5}+\frac {35 a b x \left (b x^2+a\right )^{3/2} d^4}{24 c^5}+\frac {35 a^3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right ) d^4}{16 c^5}+\frac {35 a^2 b x \sqrt {b x^2+a} d^4}{16 c^5}+\frac {\left (b x^2+a\right )^{7/2} d^3}{2 c^4 x^2}+\frac {\left (6 \left (b c^2+a d^2\right )-5 b c d x\right ) \left (b x^2+a\right )^{5/2} d^3}{30 c^6}-\frac {7 b \left (b x^2+a\right )^{5/2} d^3}{10 c^4}-\frac {7 a b \left (b x^2+a\right )^{3/2} d^3}{6 c^4}+\frac {7 a^{5/2} b \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d^3}{2 c^4}-\frac {7 a^2 b \sqrt {b x^2+a} d^3}{2 c^4}-\frac {\left (b x^2+a\right )^{7/2} d^2}{3 c^3 x^3}-\frac {7 b \left (b x^2+a\right )^{5/2} d^2}{3 c^3 x}+\frac {35 b^2 x \left (b x^2+a\right )^{3/2} d^2}{12 c^3}+\frac {35 a^2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right ) d^2}{8 c^3}+\frac {35 a b^2 x \sqrt {b x^2+a} d^2}{8 c^3}+\frac {\left (b x^2+a\right )^{7/2} d}{4 c^2 x^4}+\frac {7 b \left (b x^2+a\right )^{5/2} d}{8 c^2 x^2}+\frac {\left (8 \left (b c^2+a d^2\right )^2-b c d \left (6 b c^2+11 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2} d}{24 c^6}-\frac {35 b^2 \left (b x^2+a\right )^{3/2} d}{24 c^2}+\frac {35 a^{3/2} b^2 \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d}{8 c^2}-\frac {35 a b^2 \sqrt {b x^2+a} d}{8 c^2}-\frac {\left (b x^2+a\right )^{7/2}}{5 c x^5}-\frac {7 b \left (b x^2+a\right )^{5/2}}{15 c x^3}-\frac {7 b^2 \left (b x^2+a\right )^{3/2}}{3 c x}+\frac {7 a b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 c}+\frac {7 b^3 x \sqrt {b x^2+a}}{2 c}+\frac {\left (16 \left (b c^2+a d^2\right )^3-b c d \left (8 b^2 c^4+22 a b d^2 c^2+19 a^2 d^4\right ) x\right ) \sqrt {b x^2+a}}{16 c^6 d}-\frac {\sqrt {b} \left (16 b^3 c^6+56 a b^2 d^2 c^4+70 a^2 b d^4 c^2+35 a^3 d^6\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{16 c^5 d^2}-\frac {\left (b c^2+a d^2\right )^{7/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right )}{c^6 d^2}\)

Input:

Int[(a + b*x^2)^(7/2)/(x^6*(c + d*x)),x]
 

Output:

(-35*a*b^2*d*Sqrt[a + b*x^2])/(8*c^2) - (7*a^2*b*d^3*Sqrt[a + b*x^2])/(2*c 
^4) - (a^3*d^5*Sqrt[a + b*x^2])/c^6 + (7*b^3*x*Sqrt[a + b*x^2])/(2*c) + (3 
5*a*b^2*d^2*x*Sqrt[a + b*x^2])/(8*c^3) + (35*a^2*b*d^4*x*Sqrt[a + b*x^2])/ 
(16*c^5) + ((16*(b*c^2 + a*d^2)^3 - b*c*d*(8*b^2*c^4 + 22*a*b*c^2*d^2 + 19 
*a^2*d^4)*x)*Sqrt[a + b*x^2])/(16*c^6*d) - (35*b^2*d*(a + b*x^2)^(3/2))/(2 
4*c^2) - (7*a*b*d^3*(a + b*x^2)^(3/2))/(6*c^4) - (a^2*d^5*(a + b*x^2)^(3/2 
))/(3*c^6) - (7*b^2*(a + b*x^2)^(3/2))/(3*c*x) + (35*b^2*d^2*x*(a + b*x^2) 
^(3/2))/(12*c^3) + (35*a*b*d^4*x*(a + b*x^2)^(3/2))/(24*c^5) + (d*(8*(b*c^ 
2 + a*d^2)^2 - b*c*d*(6*b*c^2 + 11*a*d^2)*x)*(a + b*x^2)^(3/2))/(24*c^6) - 
 (7*b*d^3*(a + b*x^2)^(5/2))/(10*c^4) - (a*d^5*(a + b*x^2)^(5/2))/(5*c^6) 
- (7*b*(a + b*x^2)^(5/2))/(15*c*x^3) + (7*b*d*(a + b*x^2)^(5/2))/(8*c^2*x^ 
2) - (7*b*d^2*(a + b*x^2)^(5/2))/(3*c^3*x) + (7*b*d^4*x*(a + b*x^2)^(5/2)) 
/(6*c^5) + (d^3*(6*(b*c^2 + a*d^2) - 5*b*c*d*x)*(a + b*x^2)^(5/2))/(30*c^6 
) - (a + b*x^2)^(7/2)/(5*c*x^5) + (d*(a + b*x^2)^(7/2))/(4*c^2*x^4) - (d^2 
*(a + b*x^2)^(7/2))/(3*c^3*x^3) + (d^3*(a + b*x^2)^(7/2))/(2*c^4*x^2) - (d 
^4*(a + b*x^2)^(7/2))/(c^5*x) + (7*a*b^(5/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + 
b*x^2]])/(2*c) + (35*a^2*b^(3/2)*d^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]) 
/(8*c^3) + (35*a^3*Sqrt[b]*d^4*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*c 
^5) - (Sqrt[b]*(16*b^3*c^6 + 56*a*b^2*c^4*d^2 + 70*a^2*b*c^2*d^4 + 35*a^3* 
d^6)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*c^5*d^2) - ((b*c^2 + a*d...
 

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 = 0.52 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.22

method	result	size
risch	\(-\frac {a \sqrt {b \,x^{2}+a}\, \left (120 a^{2} d^{4} x^{4}+400 x^{4} a b \,c^{2} d^{2}+464 x^{4} b^{2} c^{4}-60 a^{2} c \,d^{3} x^{3}-195 a b \,c^{3} d \,x^{3}+40 a^{2} c^{2} d^{2} x^{2}+128 a b \,c^{4} x^{2}-30 a^{2} d \,c^{3} x +24 a^{2} c^{4}\right )}{120 c^{5} x^{5}}+\frac {-\frac {8 \left (a^{4} d^{8}+4 a^{3} b \,c^{2} d^{6}+6 a^{2} b^{2} c^{4} d^{4}+4 a \,b^{3} c^{6} d^{2}+b^{4} c^{8}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{3} c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {8 b^{4} c^{5} \left (\frac {d \sqrt {b \,x^{2}+a}}{b}-\frac {c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\right )}{d^{2}}+\frac {d \,a^{\frac {3}{2}} \left (8 a^{2} d^{4}+28 b \,c^{2} d^{2} a +35 b^{2} c^{4}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c}}{8 c^{5}}\)	\(415\)
default	\(\text {Expression too large to display}\)	\(2018\)

Input:

int((b*x^2+a)^(7/2)/x^6/(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

-1/120*a*(b*x^2+a)^(1/2)*(120*a^2*d^4*x^4+400*a*b*c^2*d^2*x^4+464*b^2*c^4* 
x^4-60*a^2*c*d^3*x^3-195*a*b*c^3*d*x^3+40*a^2*c^2*d^2*x^2+128*a*b*c^4*x^2- 
30*a^2*c^3*d*x+24*a^2*c^4)/c^5/x^5+1/8/c^5*(-8/d^3*(a^4*d^8+4*a^3*b*c^2*d^ 
6+6*a^2*b^2*c^4*d^4+4*a*b^3*c^6*d^2+b^4*c^8)/c/((a*d^2+b*c^2)/d^2)^(1/2)*l 
n((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))+8*b^4*c^5/d^2*(d* 
(b*x^2+a)^(1/2)/b-c*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2))+d*a^(3/2)*(8*a^ 
2*d^4+28*a*b*c^2*d^2+35*b^2*c^4)/c*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x^6 (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(7/2)/x^6/(d*x+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{7/2}}{x^6 (c+d x)} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {7}{2}}}{x^{6} \left (c + d x\right )}\, dx \] Input:

integrate((b*x**2+a)**(7/2)/x**6/(d*x+c),x)
 

Output:

Integral((a + b*x**2)**(7/2)/(x**6*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{7/2}}{x^6 (c+d x)} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{{\left (d x + c\right )} x^{6}} \,d x } \] Input:

integrate((b*x^2+a)^(7/2)/x^6/(d*x+c),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(7/2)/((d*x + c)*x^6), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x^6 (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*x^2+a)^(7/2)/x^6/(d*x+c),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x^6 (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{7/2}}{x^6\,\left (c+d\,x\right )} \,d x \] Input:

int((a + b*x^2)^(7/2)/(x^6*(c + d*x)),x)
 

Output:

int((a + b*x^2)^(7/2)/(x^6*(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 4129, normalized size of antiderivative = 12.14 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{x^6 (c+d x)} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^(7/2)/x^6/(d*x+c),x)
 

Output:

( - 120*sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2 
)*sqrt(a*d**2 + b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqr 
t(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**3*c*d**6*x**5 - 360* 
sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a 
*d**2 + b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqr 
t(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**2*b*c**3*d**4*x**5 - 360*sqr 
t(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d* 
*2 + b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a 
*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a*b**2*c**5*d**2*x**5 - 120*sqrt(b 
)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d**2 
+ b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d* 
*2 + b*c**2)*c - a*d**2 - 2*b*c**2))*b**3*c**7*x**5 - 120*sqrt(2*sqrt(b)*s 
qrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqr 
t(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a** 
4*d**8*x**5 - 480*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c* 
*2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b 
*c**2)*c - a*d**2 - 2*b*c**2))*a**3*b*c**2*d**6*x**5 - 720*sqrt(2*sqrt(b)* 
sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sq 
rt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a* 
*2*b**2*c**4*d**4*x**5 - 480*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a...