\(\int \frac {A+B x}{x^5 (a+b x)^{3/2}} \, dx\) [443]

Optimal result

Integrand size = 18, antiderivative size = 174 \[ \int \frac {A+B x}{x^5 (a+b x)^{3/2}} \, dx=\frac {35 b^3 (9 A b-8 a B)}{64 a^5 \sqrt {a+b x}}-\frac {A}{4 a x^4 \sqrt {a+b x}}+\frac {9 A b-8 a B}{24 a^2 x^3 \sqrt {a+b x}}-\frac {7 b (9 A b-8 a B)}{96 a^3 x^2 \sqrt {a+b x}}+\frac {35 b^2 (9 A b-8 a B)}{192 a^4 x \sqrt {a+b x}}-\frac {35 b^3 (9 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{11/2}} \]

[Out]

-35/64*b^3*(9*A*b-8*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(11/2)+35/64*b^3*(9*A*b-8*B*a)/a^5/(b*x+a)^(1/2)-1/4
*A/a/x^4/(b*x+a)^(1/2)+1/24*(9*A*b-8*B*a)/a^2/x^3/(b*x+a)^(1/2)-7/96*b*(9*A*b-8*B*a)/a^3/x^2/(b*x+a)^(1/2)+35/
192*b^2*(9*A*b-8*B*a)/a^4/x/(b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 44, 53, 65, 214} \[ \int \frac {A+B x}{x^5 (a+b x)^{3/2}} \, dx=-\frac {35 b^3 (9 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{11/2}}+\frac {35 b^3 (9 A b-8 a B)}{64 a^5 \sqrt {a+b x}}+\frac {35 b^2 (9 A b-8 a B)}{192 a^4 x \sqrt {a+b x}}-\frac {7 b (9 A b-8 a B)}{96 a^3 x^2 \sqrt {a+b x}}+\frac {9 A b-8 a B}{24 a^2 x^3 \sqrt {a+b x}}-\frac {A}{4 a x^4 \sqrt {a+b x}} \]

[In]

Int[(A + B*x)/(x^5*(a + b*x)^(3/2)),x]

[Out]

(35*b^3*(9*A*b - 8*a*B))/(64*a^5*Sqrt[a + b*x]) - A/(4*a*x^4*Sqrt[a + b*x]) + (9*A*b - 8*a*B)/(24*a^2*x^3*Sqrt
[a + b*x]) - (7*b*(9*A*b - 8*a*B))/(96*a^3*x^2*Sqrt[a + b*x]) + (35*b^2*(9*A*b - 8*a*B))/(192*a^4*x*Sqrt[a + b
*x]) - (35*b^3*(9*A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a^(11/2))

Rule 44

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

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 214

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {A}{4 a x^4 \sqrt {a+b x}}+\frac {\left (-\frac {9 A b}{2}+4 a B\right ) \int \frac {1}{x^4 (a+b x)^{3/2}} \, dx}{4 a} \\ & = -\frac {A}{4 a x^4 \sqrt {a+b x}}+\frac {9 A b-8 a B}{24 a^2 x^3 \sqrt {a+b x}}+\frac {(7 b (9 A b-8 a B)) \int \frac {1}{x^3 (a+b x)^{3/2}} \, dx}{48 a^2} \\ & = -\frac {A}{4 a x^4 \sqrt {a+b x}}+\frac {9 A b-8 a B}{24 a^2 x^3 \sqrt {a+b x}}-\frac {7 b (9 A b-8 a B)}{96 a^3 x^2 \sqrt {a+b x}}-\frac {\left (35 b^2 (9 A b-8 a B)\right ) \int \frac {1}{x^2 (a+b x)^{3/2}} \, dx}{192 a^3} \\ & = -\frac {A}{4 a x^4 \sqrt {a+b x}}+\frac {9 A b-8 a B}{24 a^2 x^3 \sqrt {a+b x}}-\frac {7 b (9 A b-8 a B)}{96 a^3 x^2 \sqrt {a+b x}}+\frac {35 b^2 (9 A b-8 a B)}{192 a^4 x \sqrt {a+b x}}+\frac {\left (35 b^3 (9 A b-8 a B)\right ) \int \frac {1}{x (a+b x)^{3/2}} \, dx}{128 a^4} \\ & = \frac {35 b^3 (9 A b-8 a B)}{64 a^5 \sqrt {a+b x}}-\frac {A}{4 a x^4 \sqrt {a+b x}}+\frac {9 A b-8 a B}{24 a^2 x^3 \sqrt {a+b x}}-\frac {7 b (9 A b-8 a B)}{96 a^3 x^2 \sqrt {a+b x}}+\frac {35 b^2 (9 A b-8 a B)}{192 a^4 x \sqrt {a+b x}}+\frac {\left (35 b^3 (9 A b-8 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{128 a^5} \\ & = \frac {35 b^3 (9 A b-8 a B)}{64 a^5 \sqrt {a+b x}}-\frac {A}{4 a x^4 \sqrt {a+b x}}+\frac {9 A b-8 a B}{24 a^2 x^3 \sqrt {a+b x}}-\frac {7 b (9 A b-8 a B)}{96 a^3 x^2 \sqrt {a+b x}}+\frac {35 b^2 (9 A b-8 a B)}{192 a^4 x \sqrt {a+b x}}+\frac {\left (35 b^2 (9 A b-8 a B)\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{64 a^5} \\ & = \frac {35 b^3 (9 A b-8 a B)}{64 a^5 \sqrt {a+b x}}-\frac {A}{4 a x^4 \sqrt {a+b x}}+\frac {9 A b-8 a B}{24 a^2 x^3 \sqrt {a+b x}}-\frac {7 b (9 A b-8 a B)}{96 a^3 x^2 \sqrt {a+b x}}+\frac {35 b^2 (9 A b-8 a B)}{192 a^4 x \sqrt {a+b x}}-\frac {35 b^3 (9 A b-8 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.75 \[ \int \frac {A+B x}{x^5 (a+b x)^{3/2}} \, dx=\frac {945 A b^4 x^4+105 a b^3 x^3 (3 A-8 B x)-16 a^4 (3 A+4 B x)+8 a^3 b x (9 A+14 B x)-14 a^2 b^2 x^2 (9 A+20 B x)}{192 a^5 x^4 \sqrt {a+b x}}+\frac {35 b^3 (-9 A b+8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{11/2}} \]

[In]

Integrate[(A + B*x)/(x^5*(a + b*x)^(3/2)),x]

[Out]

(945*A*b^4*x^4 + 105*a*b^3*x^3*(3*A - 8*B*x) - 16*a^4*(3*A + 4*B*x) + 8*a^3*b*x*(9*A + 14*B*x) - 14*a^2*b^2*x^
2*(9*A + 20*B*x))/(192*a^5*x^4*Sqrt[a + b*x]) + (35*b^3*(-9*A*b + 8*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a
^(11/2))

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.70

method	result	size
pseudoelliptic	\(\frac {-\frac {315 x^{4} \sqrt {b x +a}\, b^{3} \left (A b -\frac {8 B a}{9}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{64}+\frac {105 x^{3} \left (-\frac {8 B x}{3}+A \right ) b^{3} a^{\frac {3}{2}}}{64}-\frac {21 \left (\frac {20 B x}{9}+A \right ) x^{2} b^{2} a^{\frac {5}{2}}}{32}+\frac {3 b x \left (\frac {14 B x}{9}+A \right ) a^{\frac {7}{2}}}{8}+\frac {3 \left (-\frac {8 B x}{9}-\frac {2 A}{3}\right ) a^{\frac {9}{2}}}{8}+\frac {315 A \sqrt {a}\, b^{4} x^{4}}{64}}{a^{\frac {11}{2}} \sqrt {b x +a}\, x^{4}}\)	\(122\)
risch	\(-\frac {\sqrt {b x +a}\, \left (-561 A \,b^{3} x^{3}+456 B a \,b^{2} x^{3}+246 a A \,b^{2} x^{2}-176 B \,a^{2} b \,x^{2}-120 a^{2} A b x +64 a^{3} B x +48 a^{3} A \right )}{192 a^{5} x^{4}}+\frac {b^{3} \left (-\frac {2 \left (-128 A b +128 B a \right )}{\sqrt {b x +a}}-\frac {2 \left (315 A b -280 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{128 a^{5}}\)	\(131\)
derivativedivides	\(2 b^{3} \left (-\frac {\frac {\left (-\frac {187 A b}{128}+\frac {19 B a}{16}\right ) \left (b x +a \right )^{\frac {7}{2}}+\left (\frac {643}{128} a b A -\frac {193}{48} a^{2} B \right ) \left (b x +a \right )^{\frac {5}{2}}+\left (-\frac {765}{128} a^{2} b A +\frac {223}{48} a^{3} B \right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {325}{128} A \,a^{3} b -\frac {29}{16} B \,a^{4}\right ) \sqrt {b x +a}}{b^{4} x^{4}}+\frac {35 \left (9 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}}{a^{5}}-\frac {-A b +B a}{a^{5} \sqrt {b x +a}}\right )\)	\(148\)
default	\(2 b^{3} \left (-\frac {\frac {\left (-\frac {187 A b}{128}+\frac {19 B a}{16}\right ) \left (b x +a \right )^{\frac {7}{2}}+\left (\frac {643}{128} a b A -\frac {193}{48} a^{2} B \right ) \left (b x +a \right )^{\frac {5}{2}}+\left (-\frac {765}{128} a^{2} b A +\frac {223}{48} a^{3} B \right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {325}{128} A \,a^{3} b -\frac {29}{16} B \,a^{4}\right ) \sqrt {b x +a}}{b^{4} x^{4}}+\frac {35 \left (9 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}}{a^{5}}-\frac {-A b +B a}{a^{5} \sqrt {b x +a}}\right )\)	\(148\)

[In]

int((B*x+A)/x^5/(b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

3/8/a^(11/2)/(b*x+a)^(1/2)*(-105/8*x^4*(b*x+a)^(1/2)*b^3*(A*b-8/9*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))+35/8*x^3
*(-8/3*B*x+A)*b^3*a^(3/2)-7/4*(20/9*B*x+A)*x^2*b^2*a^(5/2)+b*x*(14/9*B*x+A)*a^(7/2)+(-8/9*B*x-2/3*A)*a^(9/2)+1
05/8*A*a^(1/2)*b^4*x^4)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.17 \[ \int \frac {A+B x}{x^5 (a+b x)^{3/2}} \, dx=\left [-\frac {105 \, {\left ({\left (8 \, B a b^{4} - 9 \, A b^{5}\right )} x^{5} + {\left (8 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4}\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (48 \, A a^{5} + 105 \, {\left (8 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 35 \, {\left (8 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} - 14 \, {\left (8 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{384 \, {\left (a^{6} b x^{5} + a^{7} x^{4}\right )}}, -\frac {105 \, {\left ({\left (8 \, B a b^{4} - 9 \, A b^{5}\right )} x^{5} + {\left (8 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (48 \, A a^{5} + 105 \, {\left (8 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 35 \, {\left (8 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} - 14 \, {\left (8 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{192 \, {\left (a^{6} b x^{5} + a^{7} x^{4}\right )}}\right ] \]

[In]

integrate((B*x+A)/x^5/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/384*(105*((8*B*a*b^4 - 9*A*b^5)*x^5 + (8*B*a^2*b^3 - 9*A*a*b^4)*x^4)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sq
rt(a) + 2*a)/x) + 2*(48*A*a^5 + 105*(8*B*a^2*b^3 - 9*A*a*b^4)*x^4 + 35*(8*B*a^3*b^2 - 9*A*a^2*b^3)*x^3 - 14*(8
*B*a^4*b - 9*A*a^3*b^2)*x^2 + 8*(8*B*a^5 - 9*A*a^4*b)*x)*sqrt(b*x + a))/(a^6*b*x^5 + a^7*x^4), -1/192*(105*((8
*B*a*b^4 - 9*A*b^5)*x^5 + (8*B*a^2*b^3 - 9*A*a*b^4)*x^4)*sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (48*A*a^5
+ 105*(8*B*a^2*b^3 - 9*A*a*b^4)*x^4 + 35*(8*B*a^3*b^2 - 9*A*a^2*b^3)*x^3 - 14*(8*B*a^4*b - 9*A*a^3*b^2)*x^2 +
8*(8*B*a^5 - 9*A*a^4*b)*x)*sqrt(b*x + a))/(a^6*b*x^5 + a^7*x^4)]

Sympy [A] (verification not implemented)

Time = 171.76 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.73 \[ \int \frac {A+B x}{x^5 (a+b x)^{3/2}} \, dx=A \left (- \frac {1}{4 a \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {3 \sqrt {b}}{8 a^{2} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {21 b^{\frac {3}{2}}}{32 a^{3} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {105 b^{\frac {5}{2}}}{64 a^{4} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {315 b^{\frac {7}{2}}}{64 a^{5} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {315 b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{64 a^{\frac {11}{2}}}\right ) + B \left (- \frac {1}{3 a \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {7 \sqrt {b}}{12 a^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {35 b^{\frac {3}{2}}}{24 a^{3} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {35 b^{\frac {5}{2}}}{8 a^{4} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {35 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 a^{\frac {9}{2}}}\right ) \]

[In]

integrate((B*x+A)/x**5/(b*x+a)**(3/2),x)

[Out]

A*(-1/(4*a*sqrt(b)*x**(9/2)*sqrt(a/(b*x) + 1)) + 3*sqrt(b)/(8*a**2*x**(7/2)*sqrt(a/(b*x) + 1)) - 21*b**(3/2)/(
32*a**3*x**(5/2)*sqrt(a/(b*x) + 1)) + 105*b**(5/2)/(64*a**4*x**(3/2)*sqrt(a/(b*x) + 1)) + 315*b**(7/2)/(64*a**
5*sqrt(x)*sqrt(a/(b*x) + 1)) - 315*b**4*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(64*a**(11/2))) + B*(-1/(3*a*sqrt(b)*
x**(7/2)*sqrt(a/(b*x) + 1)) + 7*sqrt(b)/(12*a**2*x**(5/2)*sqrt(a/(b*x) + 1)) - 35*b**(3/2)/(24*a**3*x**(3/2)*s
qrt(a/(b*x) + 1)) - 35*b**(5/2)/(8*a**4*sqrt(x)*sqrt(a/(b*x) + 1)) + 35*b**3*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/
(8*a**(9/2)))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x}{x^5 (a+b x)^{3/2}} \, dx=-\frac {1}{384} \, b^{4} {\left (\frac {2 \, {\left (384 \, B a^{5} - 384 \, A a^{4} b + 105 \, {\left (8 \, B a - 9 \, A b\right )} {\left (b x + a\right )}^{4} - 385 \, {\left (8 \, B a^{2} - 9 \, A a b\right )} {\left (b x + a\right )}^{3} + 511 \, {\left (8 \, B a^{3} - 9 \, A a^{2} b\right )} {\left (b x + a\right )}^{2} - 279 \, {\left (8 \, B a^{4} - 9 \, A a^{3} b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {9}{2}} a^{5} b - 4 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{6} b + 6 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{7} b - 4 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{8} b + \sqrt {b x + a} a^{9} b} + \frac {105 \, {\left (8 \, B a - 9 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {11}{2}} b}\right )} \]

[In]

integrate((B*x+A)/x^5/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

-1/384*b^4*(2*(384*B*a^5 - 384*A*a^4*b + 105*(8*B*a - 9*A*b)*(b*x + a)^4 - 385*(8*B*a^2 - 9*A*a*b)*(b*x + a)^3
+ 511*(8*B*a^3 - 9*A*a^2*b)*(b*x + a)^2 - 279*(8*B*a^4 - 9*A*a^3*b)*(b*x + a))/((b*x + a)^(9/2)*a^5*b - 4*(b*
x + a)^(7/2)*a^6*b + 6*(b*x + a)^(5/2)*a^7*b - 4*(b*x + a)^(3/2)*a^8*b + sqrt(b*x + a)*a^9*b) + 105*(8*B*a - 9
*A*b)*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(11/2)*b))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x}{x^5 (a+b x)^{3/2}} \, dx=-\frac {35 \, {\left (8 \, B a b^{3} - 9 \, A b^{4}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a^{5}} - \frac {2 \, {\left (B a b^{3} - A b^{4}\right )}}{\sqrt {b x + a} a^{5}} - \frac {456 \, {\left (b x + a\right )}^{\frac {7}{2}} B a b^{3} - 1544 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{2} b^{3} + 1784 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{3} b^{3} - 696 \, \sqrt {b x + a} B a^{4} b^{3} - 561 \, {\left (b x + a\right )}^{\frac {7}{2}} A b^{4} + 1929 \, {\left (b x + a\right )}^{\frac {5}{2}} A a b^{4} - 2295 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{2} b^{4} + 975 \, \sqrt {b x + a} A a^{3} b^{4}}{192 \, a^{5} b^{4} x^{4}} \]

[In]

integrate((B*x+A)/x^5/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

-35/64*(8*B*a*b^3 - 9*A*b^4)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^5) - 2*(B*a*b^3 - A*b^4)/(sqrt(b*x + a
)*a^5) - 1/192*(456*(b*x + a)^(7/2)*B*a*b^3 - 1544*(b*x + a)^(5/2)*B*a^2*b^3 + 1784*(b*x + a)^(3/2)*B*a^3*b^3
- 696*sqrt(b*x + a)*B*a^4*b^3 - 561*(b*x + a)^(7/2)*A*b^4 + 1929*(b*x + a)^(5/2)*A*a*b^4 - 2295*(b*x + a)^(3/2
)*A*a^2*b^4 + 975*sqrt(b*x + a)*A*a^3*b^4)/(a^5*b^4*x^4)

Mupad [B] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x}{x^5 (a+b x)^{3/2}} \, dx=\frac {\frac {2\,\left (A\,b^4-B\,a\,b^3\right )}{a}-\frac {93\,\left (9\,A\,b^4-8\,B\,a\,b^3\right )\,\left (a+b\,x\right )}{64\,a^2}+\frac {511\,\left (9\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^2}{192\,a^3}-\frac {385\,\left (9\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^3}{192\,a^4}+\frac {35\,\left (9\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^4}{64\,a^5}}{{\left (a+b\,x\right )}^{9/2}-4\,a\,{\left (a+b\,x\right )}^{7/2}+a^4\,\sqrt {a+b\,x}-4\,a^3\,{\left (a+b\,x\right )}^{3/2}+6\,a^2\,{\left (a+b\,x\right )}^{5/2}}-\frac {35\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (9\,A\,b-8\,B\,a\right )}{64\,a^{11/2}} \]

[In]

int((A + B*x)/(x^5*(a + b*x)^(3/2)),x)

[Out]

((2*(A*b^4 - B*a*b^3))/a - (93*(9*A*b^4 - 8*B*a*b^3)*(a + b*x))/(64*a^2) + (511*(9*A*b^4 - 8*B*a*b^3)*(a + b*x
)^2)/(192*a^3) - (385*(9*A*b^4 - 8*B*a*b^3)*(a + b*x)^3)/(192*a^4) + (35*(9*A*b^4 - 8*B*a*b^3)*(a + b*x)^4)/(6
4*a^5))/((a + b*x)^(9/2) - 4*a*(a + b*x)^(7/2) + a^4*(a + b*x)^(1/2) - 4*a^3*(a + b*x)^(3/2) + 6*a^2*(a + b*x)
^(5/2)) - (35*b^3*atanh((a + b*x)^(1/2)/a^(1/2))*(9*A*b - 8*B*a))/(64*a^(11/2))