\(\int \frac {x^{7/2} (A+B x)}{(a+b x)^3} \, dx\) [363]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.06 (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, 43, 52, 65, 211} \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^3} \, dx=\frac {7 a^{3/2} (5 A b-9 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{11/2}}-\frac {7 a \sqrt {x} (5 A b-9 a B)}{4 b^5}+\frac {7 x^{3/2} (5 A b-9 a B)}{12 b^4}-\frac {7 x^{5/2} (5 A b-9 a B)}{20 a b^3}+\frac {x^{7/2} (5 A b-9 a B)}{4 a b^2 (a+b x)}+\frac {x^{9/2} (A b-a B)}{2 a b (a+b x)^2} \]

[In]

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

[Out]

(-7*a*(5*A*b - 9*a*B)*Sqrt[x])/(4*b^5) + (7*(5*A*b - 9*a*B)*x^(3/2))/(12*b^4) - (7*(5*A*b - 9*a*B)*x^(5/2))/(2
0*a*b^3) + ((A*b - a*B)*x^(9/2))/(2*a*b*(a + b*x)^2) + ((5*A*b - 9*a*B)*x^(7/2))/(4*a*b^2*(a + b*x)) + (7*a^(3
/2)*(5*A*b - 9*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(11/2))

Rule 43

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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) x^{9/2}}{2 a b (a+b x)^2}-\frac {\left (\frac {5 A b}{2}-\frac {9 a B}{2}\right ) \int \frac {x^{7/2}}{(a+b x)^2} \, dx}{2 a b} \\ & = \frac {(A b-a B) x^{9/2}}{2 a b (a+b x)^2}+\frac {(5 A b-9 a B) x^{7/2}}{4 a b^2 (a+b x)}-\frac {(7 (5 A b-9 a B)) \int \frac {x^{5/2}}{a+b x} \, dx}{8 a b^2} \\ & = -\frac {7 (5 A b-9 a B) x^{5/2}}{20 a b^3}+\frac {(A b-a B) x^{9/2}}{2 a b (a+b x)^2}+\frac {(5 A b-9 a B) x^{7/2}}{4 a b^2 (a+b x)}+\frac {(7 (5 A b-9 a B)) \int \frac {x^{3/2}}{a+b x} \, dx}{8 b^3} \\ & = \frac {7 (5 A b-9 a B) x^{3/2}}{12 b^4}-\frac {7 (5 A b-9 a B) x^{5/2}}{20 a b^3}+\frac {(A b-a B) x^{9/2}}{2 a b (a+b x)^2}+\frac {(5 A b-9 a B) x^{7/2}}{4 a b^2 (a+b x)}-\frac {(7 a (5 A b-9 a B)) \int \frac {\sqrt {x}}{a+b x} \, dx}{8 b^4} \\ & = -\frac {7 a (5 A b-9 a B) \sqrt {x}}{4 b^5}+\frac {7 (5 A b-9 a B) x^{3/2}}{12 b^4}-\frac {7 (5 A b-9 a B) x^{5/2}}{20 a b^3}+\frac {(A b-a B) x^{9/2}}{2 a b (a+b x)^2}+\frac {(5 A b-9 a B) x^{7/2}}{4 a b^2 (a+b x)}+\frac {\left (7 a^2 (5 A b-9 a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 b^5} \\ & = -\frac {7 a (5 A b-9 a B) \sqrt {x}}{4 b^5}+\frac {7 (5 A b-9 a B) x^{3/2}}{12 b^4}-\frac {7 (5 A b-9 a B) x^{5/2}}{20 a b^3}+\frac {(A b-a B) x^{9/2}}{2 a b (a+b x)^2}+\frac {(5 A b-9 a B) x^{7/2}}{4 a b^2 (a+b x)}+\frac {\left (7 a^2 (5 A b-9 a B)\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^5} \\ & = -\frac {7 a (5 A b-9 a B) \sqrt {x}}{4 b^5}+\frac {7 (5 A b-9 a B) x^{3/2}}{12 b^4}-\frac {7 (5 A b-9 a B) x^{5/2}}{20 a b^3}+\frac {(A b-a B) x^{9/2}}{2 a b (a+b x)^2}+\frac {(5 A b-9 a B) x^{7/2}}{4 a b^2 (a+b x)}+\frac {7 a^{3/2} (5 A b-9 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.76 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^3} \, dx=\frac {\sqrt {x} \left (945 a^4 B-525 a^3 b (A-3 B x)+8 b^4 x^3 (5 A+3 B x)-8 a b^3 x^2 (35 A+9 B x)+7 a^2 b^2 x (-125 A+72 B x)\right )}{60 b^5 (a+b x)^2}-\frac {7 a^{3/2} (-5 A b+9 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{11/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {2 \left (-3 b^{2} B \,x^{2}-5 A \,b^{2} x +15 B a b x +45 a b A -90 a^{2} B \right ) \sqrt {x}}{15 b^{5}}+\frac {a^{2} \left (\frac {2 \left (-\frac {13}{8} b^{2} A +\frac {17}{8} a b B \right ) x^{\frac {3}{2}}-\frac {a \left (11 A b -15 B a \right ) \sqrt {x}}{4}}{\left (b x +a \right )^{2}}+\frac {7 \left (5 A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{b^{5}}\)	\(120\)
derivativedivides	\(-\frac {2 \left (-\frac {b^{2} B \,x^{\frac {5}{2}}}{5}-\frac {A \,b^{2} x^{\frac {3}{2}}}{3}+B a b \,x^{\frac {3}{2}}+3 a b A \sqrt {x}-6 a^{2} B \sqrt {x}\right )}{b^{5}}+\frac {2 a^{2} \left (\frac {\left (-\frac {13}{8} b^{2} A +\frac {17}{8} a b B \right ) x^{\frac {3}{2}}-\frac {a \left (11 A b -15 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {7 \left (5 A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{5}}\)	\(126\)
default	\(-\frac {2 \left (-\frac {b^{2} B \,x^{\frac {5}{2}}}{5}-\frac {A \,b^{2} x^{\frac {3}{2}}}{3}+B a b \,x^{\frac {3}{2}}+3 a b A \sqrt {x}-6 a^{2} B \sqrt {x}\right )}{b^{5}}+\frac {2 a^{2} \left (\frac {\left (-\frac {13}{8} b^{2} A +\frac {17}{8} a b B \right ) x^{\frac {3}{2}}-\frac {a \left (11 A b -15 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {7 \left (5 A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{5}}\)	\(126\)

[In]

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

[Out]

-2/15*(-3*B*b^2*x^2-5*A*b^2*x+15*B*a*b*x+45*A*a*b-90*B*a^2)*x^(1/2)/b^5+a^2/b^5*(2*((-13/8*b^2*A+17/8*a*b*B)*x
^(3/2)-1/8*a*(11*A*b-15*B*a)*x^(1/2))/(b*x+a)^2+7/4*(5*A*b-9*B*a)/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.41 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^3} \, dx=\left [-\frac {105 \, {\left (9 \, B a^{4} - 5 \, A a^{3} b + {\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 2 \, {\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (24 \, B b^{4} x^{4} + 945 \, B a^{4} - 525 \, A a^{3} b - 8 \, {\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 56 \, {\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 175 \, {\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{120 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac {105 \, {\left (9 \, B a^{4} - 5 \, A a^{3} b + {\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 2 \, {\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (24 \, B b^{4} x^{4} + 945 \, B a^{4} - 525 \, A a^{3} b - 8 \, {\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 56 \, {\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 175 \, {\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{60 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \]

[In]

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

[Out]

[-1/120*(105*(9*B*a^4 - 5*A*a^3*b + (9*B*a^2*b^2 - 5*A*a*b^3)*x^2 + 2*(9*B*a^3*b - 5*A*a^2*b^2)*x)*sqrt(-a/b)*
log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(24*B*b^4*x^4 + 945*B*a^4 - 525*A*a^3*b - 8*(9*B*a*b^3 -
5*A*b^4)*x^3 + 56*(9*B*a^2*b^2 - 5*A*a*b^3)*x^2 + 175*(9*B*a^3*b - 5*A*a^2*b^2)*x)*sqrt(x))/(b^7*x^2 + 2*a*b^
6*x + a^2*b^5), -1/60*(105*(9*B*a^4 - 5*A*a^3*b + (9*B*a^2*b^2 - 5*A*a*b^3)*x^2 + 2*(9*B*a^3*b - 5*A*a^2*b^2)*
x)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) - (24*B*b^4*x^4 + 945*B*a^4 - 525*A*a^3*b - 8*(9*B*a*b^3 - 5*A*b^4)
*x^3 + 56*(9*B*a^2*b^2 - 5*A*a*b^3)*x^2 + 175*(9*B*a^3*b - 5*A*a^2*b^2)*x)*sqrt(x))/(b^7*x^2 + 2*a*b^6*x + a^2
*b^5)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1652 vs. \(2 (163) = 326\).

Time = 58.20 (sec) , antiderivative size = 1652, normalized size of antiderivative = 9.78 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*(2*A*x**(3/2)/3 + 2*B*x**(5/2)/5), Eq(a, 0) & Eq(b, 0)), ((2*A*x**(9/2)/9 + 2*B*x**(11/2)/11)/a
**3, Eq(b, 0)), ((2*A*x**(3/2)/3 + 2*B*x**(5/2)/5)/b**3, Eq(a, 0)), (525*A*a**4*b*log(sqrt(x) - sqrt(-a/b))/(1
20*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) - 525*A*a**4*b*log(sqrt(x) + sqr
t(-a/b))/(120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) - 1050*A*a**3*b**2*sq
rt(x)*sqrt(-a/b)/(120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) + 1050*A*a**3
*b**2*x*log(sqrt(x) - sqrt(-a/b))/(120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/
b)) - 1050*A*a**3*b**2*x*log(sqrt(x) + sqrt(-a/b))/(120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b
**8*x**2*sqrt(-a/b)) - 1750*A*a**2*b**3*x**(3/2)*sqrt(-a/b)/(120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b
) + 120*b**8*x**2*sqrt(-a/b)) + 525*A*a**2*b**3*x**2*log(sqrt(x) - sqrt(-a/b))/(120*a**2*b**6*sqrt(-a/b) + 240
*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) - 525*A*a**2*b**3*x**2*log(sqrt(x) + sqrt(-a/b))/(120*a**2*b*
*6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) - 560*A*a*b**4*x**(5/2)*sqrt(-a/b)/(120*a*
*2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) + 80*A*b**5*x**(7/2)*sqrt(-a/b)/(120*
a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) - 945*B*a**5*log(sqrt(x) - sqrt(-a/
b))/(120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) + 945*B*a**5*log(sqrt(x) +
sqrt(-a/b))/(120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) + 1890*B*a**4*b*s
qrt(x)*sqrt(-a/b)/(120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) - 1890*B*a**
4*b*x*log(sqrt(x) - sqrt(-a/b))/(120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)
) + 1890*B*a**4*b*x*log(sqrt(x) + sqrt(-a/b))/(120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x
**2*sqrt(-a/b)) + 3150*B*a**3*b**2*x**(3/2)*sqrt(-a/b)/(120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 1
20*b**8*x**2*sqrt(-a/b)) - 945*B*a**3*b**2*x**2*log(sqrt(x) - sqrt(-a/b))/(120*a**2*b**6*sqrt(-a/b) + 240*a*b*
*7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) + 945*B*a**3*b**2*x**2*log(sqrt(x) + sqrt(-a/b))/(120*a**2*b**6*sq
rt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) + 1008*B*a**2*b**3*x**(5/2)*sqrt(-a/b)/(120*a**
2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) - 144*B*a*b**4*x**(7/2)*sqrt(-a/b)/(12
0*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)) + 48*B*b**5*x**(9/2)*sqrt(-a/b)/(
120*a**2*b**6*sqrt(-a/b) + 240*a*b**7*x*sqrt(-a/b) + 120*b**8*x**2*sqrt(-a/b)), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.89 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^3} \, dx=\frac {{\left (17 \, B a^{3} b - 13 \, A a^{2} b^{2}\right )} x^{\frac {3}{2}} + {\left (15 \, B a^{4} - 11 \, A a^{3} b\right )} \sqrt {x}}{4 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} - \frac {7 \, {\left (9 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{5}} + \frac {2 \, {\left (3 \, B b^{2} x^{\frac {5}{2}} - 5 \, {\left (3 \, B a b - A b^{2}\right )} x^{\frac {3}{2}} + 45 \, {\left (2 \, B a^{2} - A a b\right )} \sqrt {x}\right )}}{15 \, b^{5}} \]

[In]

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

[Out]

1/4*((17*B*a^3*b - 13*A*a^2*b^2)*x^(3/2) + (15*B*a^4 - 11*A*a^3*b)*sqrt(x))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5) -
7/4*(9*B*a^3 - 5*A*a^2*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) + 2/15*(3*B*b^2*x^(5/2) - 5*(3*B*a*b - A
*b^2)*x^(3/2) + 45*(2*B*a^2 - A*a*b)*sqrt(x))/b^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.86 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^3} \, dx=-\frac {7 \, {\left (9 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{5}} + \frac {17 \, B a^{3} b x^{\frac {3}{2}} - 13 \, A a^{2} b^{2} x^{\frac {3}{2}} + 15 \, B a^{4} \sqrt {x} - 11 \, A a^{3} b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} b^{5}} + \frac {2 \, {\left (3 \, B b^{12} x^{\frac {5}{2}} - 15 \, B a b^{11} x^{\frac {3}{2}} + 5 \, A b^{12} x^{\frac {3}{2}} + 90 \, B a^{2} b^{10} \sqrt {x} - 45 \, A a b^{11} \sqrt {x}\right )}}{15 \, b^{15}} \]

[In]

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

[Out]

-7/4*(9*B*a^3 - 5*A*a^2*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/4*(17*B*a^3*b*x^(3/2) - 13*A*a^2*b^
2*x^(3/2) + 15*B*a^4*sqrt(x) - 11*A*a^3*b*sqrt(x))/((b*x + a)^2*b^5) + 2/15*(3*B*b^12*x^(5/2) - 15*B*a*b^11*x^
(3/2) + 5*A*b^12*x^(3/2) + 90*B*a^2*b^10*sqrt(x) - 45*A*a*b^11*sqrt(x))/b^15

Mupad [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.08 \[ \int \frac {x^{7/2} (A+B x)}{(a+b x)^3} \, dx=x^{3/2}\,\left (\frac {2\,A}{3\,b^3}-\frac {2\,B\,a}{b^4}\right )-\frac {x^{3/2}\,\left (\frac {13\,A\,a^2\,b^2}{4}-\frac {17\,B\,a^3\,b}{4}\right )-\sqrt {x}\,\left (\frac {15\,B\,a^4}{4}-\frac {11\,A\,a^3\,b}{4}\right )}{a^2\,b^5+2\,a\,b^6\,x+b^7\,x^2}-\sqrt {x}\,\left (\frac {3\,a\,\left (\frac {2\,A}{b^3}-\frac {6\,B\,a}{b^4}\right )}{b}+\frac {6\,B\,a^2}{b^5}\right )+\frac {2\,B\,x^{5/2}}{5\,b^3}-\frac {7\,a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,\sqrt {x}\,\left (5\,A\,b-9\,B\,a\right )}{9\,B\,a^3-5\,A\,a^2\,b}\right )\,\left (5\,A\,b-9\,B\,a\right )}{4\,b^{11/2}} \]

[In]

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

[Out]

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