\(\int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx\) [370]

Optimal result

Integrand size = 18, antiderivative size = 169 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=-\frac {7 (9 A b-5 a B)}{20 a^3 b x^{5/2}}+\frac {7 (9 A b-5 a B)}{12 a^4 x^{3/2}}-\frac {7 b (9 A b-5 a B)}{4 a^5 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}+\frac {9 A b-5 a B}{4 a^2 b x^{5/2} (a+b x)}-\frac {7 b^{3/2} (9 A b-5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{11/2}} \]

[Out]

-7/20*(9*A*b-5*B*a)/a^3/b/x^(5/2)+7/12*(9*A*b-5*B*a)/a^4/x^(3/2)+1/2*(A*b-B*a)/a/b/x^(5/2)/(b*x+a)^2+1/4*(9*A*
b-5*B*a)/a^2/b/x^(5/2)/(b*x+a)-7/4*b^(3/2)*(9*A*b-5*B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(11/2)-7/4*b*(9*A*b
-5*B*a)/a^5/x^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 169, 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, 211} \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=-\frac {7 b^{3/2} (9 A b-5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{11/2}}-\frac {7 b (9 A b-5 a B)}{4 a^5 \sqrt {x}}+\frac {7 (9 A b-5 a B)}{12 a^4 x^{3/2}}-\frac {7 (9 A b-5 a B)}{20 a^3 b x^{5/2}}+\frac {9 A b-5 a B}{4 a^2 b x^{5/2} (a+b x)}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2} \]

[In]

Int[(A + B*x)/(x^(7/2)*(a + b*x)^3),x]

[Out]

(-7*(9*A*b - 5*a*B))/(20*a^3*b*x^(5/2)) + (7*(9*A*b - 5*a*B))/(12*a^4*x^(3/2)) - (7*b*(9*A*b - 5*a*B))/(4*a^5*
Sqrt[x]) + (A*b - a*B)/(2*a*b*x^(5/2)*(a + b*x)^2) + (9*A*b - 5*a*B)/(4*a^2*b*x^(5/2)*(a + b*x)) - (7*b^(3/2)*
(9*A*b - 5*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*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 211

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

Rubi steps \begin{align*} \text {integral}& = \frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}-\frac {\left (-\frac {9 A b}{2}+\frac {5 a B}{2}\right ) \int \frac {1}{x^{7/2} (a+b x)^2} \, dx}{2 a b} \\ & = \frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}+\frac {9 A b-5 a B}{4 a^2 b x^{5/2} (a+b x)}+\frac {(7 (9 A b-5 a B)) \int \frac {1}{x^{7/2} (a+b x)} \, dx}{8 a^2 b} \\ & = -\frac {7 (9 A b-5 a B)}{20 a^3 b x^{5/2}}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}+\frac {9 A b-5 a B}{4 a^2 b x^{5/2} (a+b x)}-\frac {(7 (9 A b-5 a B)) \int \frac {1}{x^{5/2} (a+b x)} \, dx}{8 a^3} \\ & = -\frac {7 (9 A b-5 a B)}{20 a^3 b x^{5/2}}+\frac {7 (9 A b-5 a B)}{12 a^4 x^{3/2}}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}+\frac {9 A b-5 a B}{4 a^2 b x^{5/2} (a+b x)}+\frac {(7 b (9 A b-5 a B)) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{8 a^4} \\ & = -\frac {7 (9 A b-5 a B)}{20 a^3 b x^{5/2}}+\frac {7 (9 A b-5 a B)}{12 a^4 x^{3/2}}-\frac {7 b (9 A b-5 a B)}{4 a^5 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}+\frac {9 A b-5 a B}{4 a^2 b x^{5/2} (a+b x)}-\frac {\left (7 b^2 (9 A b-5 a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a^5} \\ & = -\frac {7 (9 A b-5 a B)}{20 a^3 b x^{5/2}}+\frac {7 (9 A b-5 a B)}{12 a^4 x^{3/2}}-\frac {7 b (9 A b-5 a B)}{4 a^5 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}+\frac {9 A b-5 a B}{4 a^2 b x^{5/2} (a+b x)}-\frac {\left (7 b^2 (9 A b-5 a B)\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^5} \\ & = -\frac {7 (9 A b-5 a B)}{20 a^3 b x^{5/2}}+\frac {7 (9 A b-5 a B)}{12 a^4 x^{3/2}}-\frac {7 b (9 A b-5 a B)}{4 a^5 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} (a+b x)^2}+\frac {9 A b-5 a B}{4 a^2 b x^{5/2} (a+b x)}-\frac {7 b^{3/2} (9 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=\frac {-945 A b^4 x^4+525 a b^3 x^3 (-3 A+B x)-8 a^4 (3 A+5 B x)+8 a^3 b x (9 A+35 B x)+7 a^2 b^2 x^2 (-72 A+125 B x)}{60 a^5 x^{5/2} (a+b x)^2}+\frac {7 b^{3/2} (-9 A b+5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{11/2}} \]

[In]

Integrate[(A + B*x)/(x^(7/2)*(a + b*x)^3),x]

[Out]

(-945*A*b^4*x^4 + 525*a*b^3*x^3*(-3*A + B*x) - 8*a^4*(3*A + 5*B*x) + 8*a^3*b*x*(9*A + 35*B*x) + 7*a^2*b^2*x^2*
(-72*A + 125*B*x))/(60*a^5*x^(5/2)*(a + b*x)^2) + (7*b^(3/2)*(-9*A*b + 5*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]
])/(4*a^(11/2))

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(-\frac {2 b^{2} \left (\frac {\left (\frac {15}{8} b^{2} A -\frac {11}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (17 A b -13 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {7 \left (9 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{5}}-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 \left (-3 A b +B a \right )}{3 a^{4} x^{\frac {3}{2}}}-\frac {6 b \left (2 A b -B a \right )}{a^{5} \sqrt {x}}\)	\(121\)
default	\(-\frac {2 b^{2} \left (\frac {\left (\frac {15}{8} b^{2} A -\frac {11}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (17 A b -13 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {7 \left (9 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{5}}-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 \left (-3 A b +B a \right )}{3 a^{4} x^{\frac {3}{2}}}-\frac {6 b \left (2 A b -B a \right )}{a^{5} \sqrt {x}}\)	\(121\)
risch	\(-\frac {2 \left (90 A \,b^{2} x^{2}-45 B a b \,x^{2}-15 a A b x +5 a^{2} B x +3 a^{2} A \right )}{15 a^{5} x^{\frac {5}{2}}}-\frac {b^{2} \left (\frac {2 \left (\frac {15}{8} b^{2} A -\frac {11}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (17 A b -13 B a \right ) \sqrt {x}}{4}}{\left (b x +a \right )^{2}}+\frac {7 \left (9 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{a^{5}}\)	\(124\)

[In]

int((B*x+A)/x^(7/2)/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-2/a^5*b^2*(((15/8*b^2*A-11/8*a*b*B)*x^(3/2)+1/8*a*(17*A*b-13*B*a)*x^(1/2))/(b*x+a)^2+7/8*(9*A*b-5*B*a)/(a*b)^
(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2)))-2/5*A/a^3/x^(5/2)-2/3*(-3*A*b+B*a)/a^4/x^(3/2)-6*b*(2*A*b-B*a)/a^5/x^(1/2
)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.59 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=\left [-\frac {105 \, {\left ({\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{5} + 2 \, {\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{4} + {\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (24 \, A a^{4} - 105 \, {\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} - 175 \, {\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 56 \, {\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt {x}}{120 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}}, -\frac {105 \, {\left ({\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{5} + 2 \, {\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{4} + {\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (24 \, A a^{4} - 105 \, {\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} - 175 \, {\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 56 \, {\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt {x}}{60 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}}\right ] \]

[In]

integrate((B*x+A)/x^(7/2)/(b*x+a)^3,x, algorithm="fricas")

[Out]

[-1/120*(105*((5*B*a*b^3 - 9*A*b^4)*x^5 + 2*(5*B*a^2*b^2 - 9*A*a*b^3)*x^4 + (5*B*a^3*b - 9*A*a^2*b^2)*x^3)*sqr
t(-b/a)*log((b*x - 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 2*(24*A*a^4 - 105*(5*B*a*b^3 - 9*A*b^4)*x^4 - 175*
(5*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 56*(5*B*a^3*b - 9*A*a^2*b^2)*x^2 + 8*(5*B*a^4 - 9*A*a^3*b)*x)*sqrt(x))/(a^5*b^
2*x^5 + 2*a^6*b*x^4 + a^7*x^3), -1/60*(105*((5*B*a*b^3 - 9*A*b^4)*x^5 + 2*(5*B*a^2*b^2 - 9*A*a*b^3)*x^4 + (5*B
*a^3*b - 9*A*a^2*b^2)*x^3)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) + (24*A*a^4 - 105*(5*B*a*b^3 - 9*A*b^4)*x
^4 - 175*(5*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 56*(5*B*a^3*b - 9*A*a^2*b^2)*x^2 + 8*(5*B*a^4 - 9*A*a^3*b)*x)*sqrt(x)
)/(a^5*b^2*x^5 + 2*a^6*b*x^4 + a^7*x^3)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1882 vs. \(2 (163) = 326\).

Time = 109.15 (sec) , antiderivative size = 1882, normalized size of antiderivative = 11.14 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)/x**(7/2)/(b*x+a)**3,x)

[Out]

Piecewise((zoo*(-2*A/(11*x**(11/2)) - 2*B/(9*x**(9/2))), Eq(a, 0) & Eq(b, 0)), ((-2*A/(5*x**(5/2)) - 2*B/(3*x*
*(3/2)))/a**3, Eq(b, 0)), ((-2*A/(11*x**(11/2)) - 2*B/(9*x**(9/2)))/b**3, Eq(a, 0)), (-48*A*a**4*sqrt(-a/b)/(1
20*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) + 144*A*a**3
*b*x*sqrt(-a/b)/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-
a/b)) - 945*A*a**2*b**2*x**(5/2)*log(sqrt(x) - sqrt(-a/b))/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)
*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) + 945*A*a**2*b**2*x**(5/2)*log(sqrt(x) + sqrt(-a/b))/(120*a**
7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) - 1008*A*a**2*b**2
*x**2*sqrt(-a/b)/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(
-a/b)) - 1890*A*a*b**3*x**(7/2)*log(sqrt(x) - sqrt(-a/b))/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*
sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) + 1890*A*a*b**3*x**(7/2)*log(sqrt(x) + sqrt(-a/b))/(120*a**7*x
**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) - 3150*A*a*b**3*x**3*
sqrt(-a/b)/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b))
- 945*A*b**4*x**(9/2)*log(sqrt(x) - sqrt(-a/b))/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b
) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) + 945*A*b**4*x**(9/2)*log(sqrt(x) + sqrt(-a/b))/(120*a**7*x**(5/2)*sqrt
(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) - 1890*A*b**4*x**4*sqrt(-a/b)/(12
0*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) - 80*B*a**4*x
*sqrt(-a/b)/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)
) + 525*B*a**3*b*x**(5/2)*log(sqrt(x) - sqrt(-a/b))/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-
a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) - 525*B*a**3*b*x**(5/2)*log(sqrt(x) + sqrt(-a/b))/(120*a**7*x**(5/2)
*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) + 560*B*a**3*b*x**2*sqrt(-a/
b)/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) + 1050*
B*a**2*b**2*x**(7/2)*log(sqrt(x) - sqrt(-a/b))/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b)
+ 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) - 1050*B*a**2*b**2*x**(7/2)*log(sqrt(x) + sqrt(-a/b))/(120*a**7*x**(5/2)*
sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) + 1750*B*a**2*b**2*x**3*sqrt(
-a/b)/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) + 52
5*B*a*b**3*x**(9/2)*log(sqrt(x) - sqrt(-a/b))/(120*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) +
120*a**5*b**2*x**(9/2)*sqrt(-a/b)) - 525*B*a*b**3*x**(9/2)*log(sqrt(x) + sqrt(-a/b))/(120*a**7*x**(5/2)*sqrt(
-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)) + 1050*B*a*b**3*x**4*sqrt(-a/b)/(1
20*a**7*x**(5/2)*sqrt(-a/b) + 240*a**6*b*x**(7/2)*sqrt(-a/b) + 120*a**5*b**2*x**(9/2)*sqrt(-a/b)), True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=-\frac {24 \, A a^{4} - 105 \, {\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} - 175 \, {\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 56 \, {\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x}{60 \, {\left (a^{5} b^{2} x^{\frac {9}{2}} + 2 \, a^{6} b x^{\frac {7}{2}} + a^{7} x^{\frac {5}{2}}\right )}} + \frac {7 \, {\left (5 \, B a b^{2} - 9 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{5}} \]

[In]

integrate((B*x+A)/x^(7/2)/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/60*(24*A*a^4 - 105*(5*B*a*b^3 - 9*A*b^4)*x^4 - 175*(5*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 56*(5*B*a^3*b - 9*A*a^2*
b^2)*x^2 + 8*(5*B*a^4 - 9*A*a^3*b)*x)/(a^5*b^2*x^(9/2) + 2*a^6*b*x^(7/2) + a^7*x^(5/2)) + 7/4*(5*B*a*b^2 - 9*A
*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.80 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=\frac {7 \, {\left (5 \, B a b^{2} - 9 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{5}} + \frac {11 \, B a b^{3} x^{\frac {3}{2}} - 15 \, A b^{4} x^{\frac {3}{2}} + 13 \, B a^{2} b^{2} \sqrt {x} - 17 \, A a b^{3} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{5}} + \frac {2 \, {\left (45 \, B a b x^{2} - 90 \, A b^{2} x^{2} - 5 \, B a^{2} x + 15 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{5} x^{\frac {5}{2}}} \]

[In]

integrate((B*x+A)/x^(7/2)/(b*x+a)^3,x, algorithm="giac")

[Out]

7/4*(5*B*a*b^2 - 9*A*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) + 1/4*(11*B*a*b^3*x^(3/2) - 15*A*b^4*x^(
3/2) + 13*B*a^2*b^2*sqrt(x) - 17*A*a*b^3*sqrt(x))/((b*x + a)^2*a^5) + 2/15*(45*B*a*b*x^2 - 90*A*b^2*x^2 - 5*B*
a^2*x + 15*A*a*b*x - 3*A*a^2)/(a^5*x^(5/2))

Mupad [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.80 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^3} \, dx=-\frac {\frac {2\,A}{5\,a}-\frac {2\,x\,\left (9\,A\,b-5\,B\,a\right )}{15\,a^2}+\frac {35\,b^2\,x^3\,\left (9\,A\,b-5\,B\,a\right )}{12\,a^4}+\frac {7\,b^3\,x^4\,\left (9\,A\,b-5\,B\,a\right )}{4\,a^5}+\frac {14\,b\,x^2\,\left (9\,A\,b-5\,B\,a\right )}{15\,a^3}}{a^2\,x^{5/2}+b^2\,x^{9/2}+2\,a\,b\,x^{7/2}}-\frac {7\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (9\,A\,b-5\,B\,a\right )}{4\,a^{11/2}} \]

[In]

int((A + B*x)/(x^(7/2)*(a + b*x)^3),x)

[Out]

- ((2*A)/(5*a) - (2*x*(9*A*b - 5*B*a))/(15*a^2) + (35*b^2*x^3*(9*A*b - 5*B*a))/(12*a^4) + (7*b^3*x^4*(9*A*b -
5*B*a))/(4*a^5) + (14*b*x^2*(9*A*b - 5*B*a))/(15*a^3))/(a^2*x^(5/2) + b^2*x^(9/2) + 2*a*b*x^(7/2)) - (7*b^(3/2
)*atan((b^(1/2)*x^(1/2))/a^(1/2))*(9*A*b - 5*B*a))/(4*a^(11/2))