\(\int \frac {(A+B x) (a+b x^2)^{5/2}}{x^7} \, dx\) [1124]

Optimal result

Integrand size = 20, antiderivative size = 140 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{5/2}}{x^7} \, dx=-\frac {b^2 (5 A+16 B x) \sqrt {a+b x^2}}{16 x^2}-\frac {b (5 A+8 B x) \left (a+b x^2\right )^{3/2}}{24 x^4}-\frac {(5 A+6 B x) \left (a+b x^2\right )^{5/2}}{30 x^6}+b^{5/2} B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {5 A b^3 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}} \] Output:

-1/16*b^2*(16*B*x+5*A)*(b*x^2+a)^(1/2)/x^2-1/24*b*(8*B*x+5*A)*(b*x^2+a)^(3 
/2)/x^4-1/30*(6*B*x+5*A)*(b*x^2+a)^(5/2)/x^6+b^(5/2)*B*arctanh(b^(1/2)*x/( 
b*x^2+a)^(1/2))-5/16*A*b^3*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(1/2)
 

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{5/2}}{x^7} \, dx=-\frac {\sqrt {a+b x^2} \left (8 a^2 (5 A+6 B x)+2 a b x^2 (65 A+88 B x)+b^2 x^4 (165 A+368 B x)\right )}{240 x^6}+\frac {5 A b^3 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}}-b^{5/2} B \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \] Input:

Integrate[((A + B*x)*(a + b*x^2)^(5/2))/x^7,x]
 

Output:

-1/240*(Sqrt[a + b*x^2]*(8*a^2*(5*A + 6*B*x) + 2*a*b*x^2*(65*A + 88*B*x) + 
 b^2*x^4*(165*A + 368*B*x)))/x^6 + (5*A*b^3*ArcTanh[(Sqrt[b]*x - Sqrt[a + 
b*x^2])/Sqrt[a]])/(8*Sqrt[a]) - b^(5/2)*B*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^ 
2]]
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {537, 25, 537, 27, 537, 25, 538, 224, 219, 243, 73, 221}

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} (A+B x)}{x^7} \, dx\)

\(\Big \downarrow \) 537

\(\displaystyle -\frac {1}{6} b \int -\frac {(5 A+6 B x) \left (b x^2+a\right )^{3/2}}{x^5}dx-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} b \int \frac {(5 A+6 B x) \left (b x^2+a\right )^{3/2}}{x^5}dx-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

\(\Big \downarrow \) 537

\(\displaystyle \frac {1}{6} b \left (-\frac {1}{4} b \int -\frac {3 (5 A+8 B x) \sqrt {b x^2+a}}{x^3}dx-\frac {\left (a+b x^2\right )^{3/2} (5 A+8 B x)}{4 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} b \left (\frac {3}{4} b \int \frac {(5 A+8 B x) \sqrt {b x^2+a}}{x^3}dx-\frac {\left (a+b x^2\right )^{3/2} (5 A+8 B x)}{4 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

\(\Big \downarrow \) 537

\(\displaystyle \frac {1}{6} b \left (\frac {3}{4} b \left (-\frac {1}{2} b \int -\frac {5 A+16 B x}{x \sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} (5 A+16 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (5 A+8 B x)}{4 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} b \left (\frac {3}{4} b \left (\frac {1}{2} b \int \frac {5 A+16 B x}{x \sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} (5 A+16 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (5 A+8 B x)}{4 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {1}{6} b \left (\frac {3}{4} b \left (\frac {1}{2} b \left (5 A \int \frac {1}{x \sqrt {b x^2+a}}dx+16 B \int \frac {1}{\sqrt {b x^2+a}}dx\right )-\frac {\sqrt {a+b x^2} (5 A+16 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (5 A+8 B x)}{4 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {1}{6} b \left (\frac {3}{4} b \left (\frac {1}{2} b \left (5 A \int \frac {1}{x \sqrt {b x^2+a}}dx+16 B \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}\right )-\frac {\sqrt {a+b x^2} (5 A+16 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (5 A+8 B x)}{4 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{6} b \left (\frac {3}{4} b \left (\frac {1}{2} b \left (5 A \int \frac {1}{x \sqrt {b x^2+a}}dx+\frac {16 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} (5 A+16 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (5 A+8 B x)}{4 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{6} b \left (\frac {3}{4} b \left (\frac {1}{2} b \left (\frac {5}{2} A \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2+\frac {16 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} (5 A+16 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (5 A+8 B x)}{4 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{6} b \left (\frac {3}{4} b \left (\frac {1}{2} b \left (\frac {5 A \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}+\frac {16 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} (5 A+16 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (5 A+8 B x)}{4 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{6} b \left (\frac {3}{4} b \left (\frac {1}{2} b \left (\frac {16 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}-\frac {5 A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right )-\frac {\sqrt {a+b x^2} (5 A+16 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (5 A+8 B x)}{4 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}\)

Input:

Int[((A + B*x)*(a + b*x^2)^(5/2))/x^7,x]
 

Output:

-1/30*((5*A + 6*B*x)*(a + b*x^2)^(5/2))/x^6 + (b*(-1/4*((5*A + 8*B*x)*(a + 
 b*x^2)^(3/2))/x^4 + (3*b*(-1/2*((5*A + 16*B*x)*Sqrt[a + b*x^2])/x^2 + (b* 
((16*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - (5*A*ArcTanh[Sqrt[a 
 + b*x^2]/Sqrt[a]])/Sqrt[a]))/2))/4))/6
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 537

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), 
 x] - Simp[2*b*(p/((m + 1)*(m + 2)))   Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) 
*x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && 
 GtQ[p, 0] &&  !ILtQ[m + 2*p + 3, 0] && IntegerQ[2*p]
 

rule 538

Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp 
[c   Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d   Int[1/Sqrt[a + b*x^2], x] 
, x] /; FreeQ[{a, b, c, d}, x]
 

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.84

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (368 B \,b^{2} x^{5}+165 b^{2} A \,x^{4}+176 a b B \,x^{3}+130 a b A \,x^{2}+48 a^{2} B x +40 a^{2} A \right )}{240 x^{6}}+b^{\frac {5}{2}} B \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )-\frac {5 b^{3} A \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{16 \sqrt {a}}\)	\(117\)
default	\(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{6 a \,x^{6}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}}+\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}}+\frac {5 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{5 a \,x^{5}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{3 a \,x^{3}}+\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{a x}+\frac {6 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{a}\right )}{3 a}\right )}{5 a}\right )\)	\(283\)

Input:

int((B*x+A)*(b*x^2+a)^(5/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/240*(b*x^2+a)^(1/2)*(368*B*b^2*x^5+165*A*b^2*x^4+176*B*a*b*x^3+130*A*a* 
b*x^2+48*B*a^2*x+40*A*a^2)/x^6+b^(5/2)*B*ln(b^(1/2)*x+(b*x^2+a)^(1/2))-5/1 
6*b^3*A/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 580, normalized size of antiderivative = 4.14 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{5/2}}{x^7} \, dx=\left [\frac {240 \, B a b^{\frac {5}{2}} x^{6} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 75 \, A \sqrt {a} b^{3} x^{6} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (368 \, B a b^{2} x^{5} + 165 \, A a b^{2} x^{4} + 176 \, B a^{2} b x^{3} + 130 \, A a^{2} b x^{2} + 48 \, B a^{3} x + 40 \, A a^{3}\right )} \sqrt {b x^{2} + a}}{480 \, a x^{6}}, -\frac {480 \, B a \sqrt {-b} b^{2} x^{6} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 75 \, A \sqrt {a} b^{3} x^{6} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (368 \, B a b^{2} x^{5} + 165 \, A a b^{2} x^{4} + 176 \, B a^{2} b x^{3} + 130 \, A a^{2} b x^{2} + 48 \, B a^{3} x + 40 \, A a^{3}\right )} \sqrt {b x^{2} + a}}{480 \, a x^{6}}, \frac {75 \, A \sqrt {-a} b^{3} x^{6} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + 120 \, B a b^{\frac {5}{2}} x^{6} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - {\left (368 \, B a b^{2} x^{5} + 165 \, A a b^{2} x^{4} + 176 \, B a^{2} b x^{3} + 130 \, A a^{2} b x^{2} + 48 \, B a^{3} x + 40 \, A a^{3}\right )} \sqrt {b x^{2} + a}}{240 \, a x^{6}}, -\frac {240 \, B a \sqrt {-b} b^{2} x^{6} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 75 \, A \sqrt {-a} b^{3} x^{6} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (368 \, B a b^{2} x^{5} + 165 \, A a b^{2} x^{4} + 176 \, B a^{2} b x^{3} + 130 \, A a^{2} b x^{2} + 48 \, B a^{3} x + 40 \, A a^{3}\right )} \sqrt {b x^{2} + a}}{240 \, a x^{6}}\right ] \] Input:

integrate((B*x+A)*(b*x^2+a)^(5/2)/x^7,x, algorithm="fricas")
 

Output:

[1/480*(240*B*a*b^(5/2)*x^6*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a 
) + 75*A*sqrt(a)*b^3*x^6*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^ 
2) - 2*(368*B*a*b^2*x^5 + 165*A*a*b^2*x^4 + 176*B*a^2*b*x^3 + 130*A*a^2*b* 
x^2 + 48*B*a^3*x + 40*A*a^3)*sqrt(b*x^2 + a))/(a*x^6), -1/480*(480*B*a*sqr 
t(-b)*b^2*x^6*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 75*A*sqrt(a)*b^3*x^6*lo 
g(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(368*B*a*b^2*x^5 + 1 
65*A*a*b^2*x^4 + 176*B*a^2*b*x^3 + 130*A*a^2*b*x^2 + 48*B*a^3*x + 40*A*a^3 
)*sqrt(b*x^2 + a))/(a*x^6), 1/240*(75*A*sqrt(-a)*b^3*x^6*arctan(sqrt(b*x^2 
 + a)*sqrt(-a)/a) + 120*B*a*b^(5/2)*x^6*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*s 
qrt(b)*x - a) - (368*B*a*b^2*x^5 + 165*A*a*b^2*x^4 + 176*B*a^2*b*x^3 + 130 
*A*a^2*b*x^2 + 48*B*a^3*x + 40*A*a^3)*sqrt(b*x^2 + a))/(a*x^6), -1/240*(24 
0*B*a*sqrt(-b)*b^2*x^6*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 75*A*sqrt(-a)* 
b^3*x^6*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (368*B*a*b^2*x^5 + 165*A*a*b^ 
2*x^4 + 176*B*a^2*b*x^3 + 130*A*a^2*b*x^2 + 48*B*a^3*x + 40*A*a^3)*sqrt(b* 
x^2 + a))/(a*x^6)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (129) = 258\).

Time = 9.33 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.14 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{5/2}}{x^7} \, dx=- \frac {A a^{3}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 A a^{2} \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {35 A a b^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {3 A b^{\frac {5}{2}}}{16 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 A b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 \sqrt {a}} - \frac {B \sqrt {a} b^{2}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{2} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {11 B a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 x^{2}} - \frac {8 B b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15} + B b^{\frac {5}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {B b^{3} x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:

integrate((B*x+A)*(b*x**2+a)**(5/2)/x**7,x)
 

Output:

-A*a**3/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 17*A*a**2*sqrt(b)/(24*x**5 
*sqrt(a/(b*x**2) + 1)) - 35*A*a*b**(3/2)/(48*x**3*sqrt(a/(b*x**2) + 1)) - 
A*b**(5/2)*sqrt(a/(b*x**2) + 1)/(2*x) - 3*A*b**(5/2)/(16*x*sqrt(a/(b*x**2) 
 + 1)) - 5*A*b**3*asinh(sqrt(a)/(sqrt(b)*x))/(16*sqrt(a)) - B*sqrt(a)*b**2 
/(x*sqrt(1 + b*x**2/a)) - B*a**2*sqrt(b)*sqrt(a/(b*x**2) + 1)/(5*x**4) - 1 
1*B*a*b**(3/2)*sqrt(a/(b*x**2) + 1)/(15*x**2) - 8*B*b**(5/2)*sqrt(a/(b*x** 
2) + 1)/15 + B*b**(5/2)*asinh(sqrt(b)*x/sqrt(a)) - B*b**3*x/(sqrt(a)*sqrt( 
1 + b*x**2/a))
                                                                                    
                                                                                    
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (114) = 228\).

Time = 0.04 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.74 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{5/2}}{x^7} \, dx=\frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{3} x}{3 \, a^{2}} + \frac {\sqrt {b x^{2} + a} B b^{3} x}{a} + B b^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {5 \, A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, \sqrt {a}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3}}{16 \, a^{3}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{48 \, a^{2}} + \frac {5 \, \sqrt {b x^{2} + a} A b^{3}}{16 \, a} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{15 \, a^{2} x} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{2}}{16 \, a^{3} x^{2}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B b}{15 \, a^{2} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A b}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B}{5 \, a x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{6 \, a x^{6}} \] Input:

integrate((B*x+A)*(b*x^2+a)^(5/2)/x^7,x, algorithm="maxima")
 

Output:

2/3*(b*x^2 + a)^(3/2)*B*b^3*x/a^2 + sqrt(b*x^2 + a)*B*b^3*x/a + B*b^(5/2)* 
arcsinh(b*x/sqrt(a*b)) - 5/16*A*b^3*arcsinh(a/(sqrt(a*b)*abs(x)))/sqrt(a) 
+ 1/16*(b*x^2 + a)^(5/2)*A*b^3/a^3 + 5/48*(b*x^2 + a)^(3/2)*A*b^3/a^2 + 5/ 
16*sqrt(b*x^2 + a)*A*b^3/a - 8/15*(b*x^2 + a)^(5/2)*B*b^2/(a^2*x) - 1/16*( 
b*x^2 + a)^(7/2)*A*b^2/(a^3*x^2) - 2/15*(b*x^2 + a)^(7/2)*B*b/(a^2*x^3) - 
1/24*(b*x^2 + a)^(7/2)*A*b/(a^2*x^4) - 1/5*(b*x^2 + a)^(7/2)*B/(a*x^5) - 1 
/6*(b*x^2 + a)^(7/2)*A/(a*x^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (114) = 228\).

Time = 0.16 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.84 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{5/2}}{x^7} \, dx=\frac {5 \, A b^{3} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a}} - B b^{\frac {5}{2}} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \frac {165 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{11} A b^{3} + 720 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a b^{\frac {5}{2}} + 25 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} A a b^{3} - 2160 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{2} b^{\frac {5}{2}} + 450 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} A a^{2} b^{3} + 3680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{3} b^{\frac {5}{2}} + 450 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} A a^{3} b^{3} - 3360 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{4} b^{\frac {5}{2}} + 25 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A a^{4} b^{3} + 1488 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{5} b^{\frac {5}{2}} + 165 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{5} b^{3} - 368 \, B a^{6} b^{\frac {5}{2}}}{120 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{6}} \] Input:

integrate((B*x+A)*(b*x^2+a)^(5/2)/x^7,x, algorithm="giac")
 

Output:

5/8*A*b^3*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/sqrt(-a) - B*b^( 
5/2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a))) + 1/120*(165*(sqrt(b)*x - sqrt 
(b*x^2 + a))^11*A*b^3 + 720*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a*b^(5/2) + 
 25*(sqrt(b)*x - sqrt(b*x^2 + a))^9*A*a*b^3 - 2160*(sqrt(b)*x - sqrt(b*x^2 
 + a))^8*B*a^2*b^(5/2) + 450*(sqrt(b)*x - sqrt(b*x^2 + a))^7*A*a^2*b^3 + 3 
680*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^3*b^(5/2) + 450*(sqrt(b)*x - sqrt( 
b*x^2 + a))^5*A*a^3*b^3 - 3360*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^4*b^(5/ 
2) + 25*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a^4*b^3 + 1488*(sqrt(b)*x - sqrt 
(b*x^2 + a))^2*B*a^5*b^(5/2) + 165*(sqrt(b)*x - sqrt(b*x^2 + a))*A*a^5*b^3 
 - 368*B*a^6*b^(5/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^6
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x^2\right )^{5/2}}{x^7} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )}{x^7} \,d x \] Input:

int(((a + b*x^2)^(5/2)*(A + B*x))/x^7,x)
 

Output:

int(((a + b*x^2)^(5/2)*(A + B*x))/x^7, x)
 

Reduce [F]

\[ \int \frac {(A+B x) \left (a+b x^2\right )^{5/2}}{x^7} \, dx=\int \frac {\left (B x +A \right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{x^{7}}d x \] Input:

int((B*x+A)*(b*x^2+a)^(5/2)/x^7,x)
 

Output:

int((B*x+A)*(b*x^2+a)^(5/2)/x^7,x)