3.362 \(\int \frac {A+B x}{x^{9/2} (a+b x)^2} \, dx\)

Optimal. Leaf size=153 \[ \frac {b^{5/2} (9 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{11/2}}+\frac {b^2 (9 A b-7 a B)}{a^5 \sqrt {x}}-\frac {b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac {9 A b-7 a B}{5 a^3 x^{5/2}}-\frac {9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac {A b-a B}{a b x^{7/2} (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {78, 51, 63, 205} \[ \frac {b^2 (9 A b-7 a B)}{a^5 \sqrt {x}}+\frac {b^{5/2} (9 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{11/2}}-\frac {b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac {9 A b-7 a B}{5 a^3 x^{5/2}}-\frac {9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac {A b-a B}{a b x^{7/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(9/2)*(a + b*x)^2),x]

[Out]

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

Rule 51

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 63

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 78

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] || LtQ
[p, n]))))

Rule 205

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

Rubi steps

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

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Mathematica [C]  time = 0.02, size = 64, normalized size = 0.42 \[ \frac {(a+b x) (7 a B-9 A b) \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};-\frac {b x}{a}\right )+7 a (A b-a B)}{7 a^2 b x^{7/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(9/2)*(a + b*x)^2),x]

[Out]

(7*a*(A*b - a*B) + (-9*A*b + 7*a*B)*(a + b*x)*Hypergeometric2F1[-7/2, 1, -5/2, -((b*x)/a)])/(7*a^2*b*x^(7/2)*(
a + b*x))

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fricas [A]  time = 0.61, size = 372, normalized size = 2.43 \[ \left [-\frac {105 \, {\left ({\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{5} + {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{4}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (30 \, A a^{4} + 105 \, {\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 70 \, {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 14 \, {\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 6 \, {\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt {x}}{210 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}}, \frac {105 \, {\left ({\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{5} + {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{4}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (30 \, A a^{4} + 105 \, {\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 70 \, {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 14 \, {\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 6 \, {\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt {x}}{105 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(9/2)/(b*x+a)^2,x, algorithm="fricas")

[Out]

[-1/210*(105*((7*B*a*b^3 - 9*A*b^4)*x^5 + (7*B*a^2*b^2 - 9*A*a*b^3)*x^4)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqr
t(-b/a) - a)/(b*x + a)) + 2*(30*A*a^4 + 105*(7*B*a*b^3 - 9*A*b^4)*x^4 + 70*(7*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 14*
(7*B*a^3*b - 9*A*a^2*b^2)*x^2 + 6*(7*B*a^4 - 9*A*a^3*b)*x)*sqrt(x))/(a^5*b*x^5 + a^6*x^4), 1/105*(105*((7*B*a*
b^3 - 9*A*b^4)*x^5 + (7*B*a^2*b^2 - 9*A*a*b^3)*x^4)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) - (30*A*a^4 + 10
5*(7*B*a*b^3 - 9*A*b^4)*x^4 + 70*(7*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 14*(7*B*a^3*b - 9*A*a^2*b^2)*x^2 + 6*(7*B*a^4
- 9*A*a^3*b)*x)*sqrt(x))/(a^5*b*x^5 + a^6*x^4)]

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giac [A]  time = 1.26, size = 136, normalized size = 0.89 \[ -\frac {{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} - \frac {B a b^{3} \sqrt {x} - A b^{4} \sqrt {x}}{{\left (b x + a\right )} a^{5}} - \frac {2 \, {\left (315 \, B a b^{2} x^{3} - 420 \, A b^{3} x^{3} - 70 \, B a^{2} b x^{2} + 105 \, A a b^{2} x^{2} + 21 \, B a^{3} x - 42 \, A a^{2} b x + 15 \, A a^{3}\right )}}{105 \, a^{5} x^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(9/2)/(b*x+a)^2,x, algorithm="giac")

[Out]

-(7*B*a*b^3 - 9*A*b^4)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) - (B*a*b^3*sqrt(x) - A*b^4*sqrt(x))/((b*x +
a)*a^5) - 2/105*(315*B*a*b^2*x^3 - 420*A*b^3*x^3 - 70*B*a^2*b*x^2 + 105*A*a*b^2*x^2 + 21*B*a^3*x - 42*A*a^2*b
*x + 15*A*a^3)/(a^5*x^(7/2))

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maple [A]  time = 0.02, size = 163, normalized size = 1.07 \[ \frac {9 A \,b^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{5}}-\frac {7 B \,b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{4}}+\frac {A \,b^{4} \sqrt {x}}{\left (b x +a \right ) a^{5}}-\frac {B \,b^{3} \sqrt {x}}{\left (b x +a \right ) a^{4}}+\frac {8 A \,b^{3}}{a^{5} \sqrt {x}}-\frac {6 B \,b^{2}}{a^{4} \sqrt {x}}-\frac {2 A \,b^{2}}{a^{4} x^{\frac {3}{2}}}+\frac {4 B b}{3 a^{3} x^{\frac {3}{2}}}+\frac {4 A b}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 B}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 A}{7 a^{2} x^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(9/2)/(b*x+a)^2,x)

[Out]

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

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maxima [A]  time = 1.99, size = 143, normalized size = 0.93 \[ -\frac {30 \, A a^{4} + 105 \, {\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 70 \, {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 14 \, {\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 6 \, {\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x}{105 \, {\left (a^{5} b x^{\frac {9}{2}} + a^{6} x^{\frac {7}{2}}\right )}} - \frac {{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(9/2)/(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/105*(30*A*a^4 + 105*(7*B*a*b^3 - 9*A*b^4)*x^4 + 70*(7*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 14*(7*B*a^3*b - 9*A*a^2*
b^2)*x^2 + 6*(7*B*a^4 - 9*A*a^3*b)*x)/(a^5*b*x^(9/2) + a^6*x^(7/2)) - (7*B*a*b^3 - 9*A*b^4)*arctan(b*sqrt(x)/s
qrt(a*b))/(sqrt(a*b)*a^5)

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mupad [B]  time = 0.46, size = 121, normalized size = 0.79 \[ \frac {\frac {2\,x\,\left (9\,A\,b-7\,B\,a\right )}{35\,a^2}-\frac {2\,A}{7\,a}+\frac {2\,b^2\,x^3\,\left (9\,A\,b-7\,B\,a\right )}{3\,a^4}+\frac {b^3\,x^4\,\left (9\,A\,b-7\,B\,a\right )}{a^5}-\frac {2\,b\,x^2\,\left (9\,A\,b-7\,B\,a\right )}{15\,a^3}}{a\,x^{7/2}+b\,x^{9/2}}+\frac {b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (9\,A\,b-7\,B\,a\right )}{a^{11/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(9/2)*(a + b*x)^2),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**(9/2)/(b*x+a)**2,x)

[Out]

Timed out

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