3.4.52 \(\int \frac {A+B x}{x^{9/2} (a+b x)} \, dx\) [352]

Optimal. Leaf size=113 \[ -\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 b (A b-a B)}{3 a^3 x^{3/2}}+\frac {2 b^2 (A b-a B)}{a^4 \sqrt {x}}+\frac {2 b^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 53, 65, 211} \begin {gather*} \frac {2 b^{5/2} (A b-a B) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {2 b^2 (A b-a B)}{a^4 \sqrt {x}}-\frac {2 b (A b-a B)}{3 a^3 x^{3/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 A}{7 a x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(9/2)*(a + b*x)),x]

[Out]

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

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

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Mathematica [A]
time = 0.14, size = 103, normalized size = 0.91 \begin {gather*} \frac {210 A b^3 x^3-70 a b^2 x^2 (A+3 B x)+14 a^2 b x (3 A+5 B x)-6 a^3 (5 A+7 B x)}{105 a^4 x^{7/2}}+\frac {2 b^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(210*A*b^3*x^3 - 70*a*b^2*x^2*(A + 3*B*x) + 14*a^2*b*x*(3*A + 5*B*x) - 6*a^3*(5*A + 7*B*x))/(105*a^4*x^(7/2))
+ (2*b^(5/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(9/2)

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Maple [A]
time = 0.07, size = 95, normalized size = 0.84

method	result	size
derivativedivides	\(-\frac {2 A}{7 a \,x^{\frac {7}{2}}}-\frac {2 \left (-A b +B a \right )}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 b \left (A b -B a \right )}{3 a^{3} x^{\frac {3}{2}}}+\frac {2 b^{2} \left (A b -B a \right )}{a^{4} \sqrt {x}}+\frac {2 \left (A b -B a \right ) b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a^{4} \sqrt {a b}}\)	\(95\)
default	\(-\frac {2 A}{7 a \,x^{\frac {7}{2}}}-\frac {2 \left (-A b +B a \right )}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 b \left (A b -B a \right )}{3 a^{3} x^{\frac {3}{2}}}+\frac {2 b^{2} \left (A b -B a \right )}{a^{4} \sqrt {x}}+\frac {2 \left (A b -B a \right ) b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a^{4} \sqrt {a b}}\)	\(95\)
risch	\(-\frac {2 \left (-105 A \,b^{3} x^{3}+105 B a \,b^{2} x^{3}+35 a A \,b^{2} x^{2}-35 B \,a^{2} b \,x^{2}-21 a^{2} A b x +21 a^{3} B x +15 a^{3} A \right )}{105 a^{4} x^{\frac {7}{2}}}+\frac {2 b^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) A}{a^{4} \sqrt {a b}}-\frac {2 b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) B}{a^{3} \sqrt {a b}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 103, normalized size = 0.91 \begin {gather*} -\frac {2 \, {\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {2 \, {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )}}{105 \, a^{4} x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(B*a*b^3 - A*b^4)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) - 2/105*(15*A*a^3 + 105*(B*a*b^2 - A*b^3)*x^3
- 35*(B*a^2*b - A*a*b^2)*x^2 + 21*(B*a^3 - A*a^2*b)*x)/(a^4*x^(7/2))

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Fricas [A]
time = 1.10, size = 246, normalized size = 2.18 \begin {gather*} \left [-\frac {105 \, {\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt {x}}{105 \, a^{4} x^{4}}, \frac {2 \, {\left (105 \, {\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (15 \, A a^{3} + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt {x}\right )}}{105 \, a^{4} x^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/105*(105*(B*a*b^2 - A*b^3)*x^4*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 2*(15*A*a^3
+ 105*(B*a*b^2 - A*b^3)*x^3 - 35*(B*a^2*b - A*a*b^2)*x^2 + 21*(B*a^3 - A*a^2*b)*x)*sqrt(x))/(a^4*x^4), 2/105*(
105*(B*a*b^2 - A*b^3)*x^4*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) - (15*A*a^3 + 105*(B*a*b^2 - A*b^3)*x^3 -
35*(B*a^2*b - A*a*b^2)*x^2 + 21*(B*a^3 - A*a^2*b)*x)*sqrt(x))/(a^4*x^4)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (109) = 218\).
time = 34.64, size = 299, normalized size = 2.65 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{b} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}}{a} & \text {for}\: b = 0 \\- \frac {2 A}{7 a x^{\frac {7}{2}}} + \frac {2 A b}{5 a^{2} x^{\frac {5}{2}}} - \frac {2 A b^{2}}{3 a^{3} x^{\frac {3}{2}}} + \frac {A b^{3} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{a^{4} \sqrt {- \frac {a}{b}}} - \frac {A b^{3} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{a^{4} \sqrt {- \frac {a}{b}}} + \frac {2 A b^{3}}{a^{4} \sqrt {x}} - \frac {2 B}{5 a x^{\frac {5}{2}}} + \frac {2 B b}{3 a^{2} x^{\frac {3}{2}}} - \frac {B b^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{a^{3} \sqrt {- \frac {a}{b}}} + \frac {B b^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{a^{3} \sqrt {- \frac {a}{b}}} - \frac {2 B b^{2}}{a^{3} \sqrt {x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2))), Eq(a, 0) & Eq(b, 0)), ((-2*A/(9*x**(9/2)) - 2*B/(7*x**(
7/2)))/b, Eq(a, 0)), ((-2*A/(7*x**(7/2)) - 2*B/(5*x**(5/2)))/a, Eq(b, 0)), (-2*A/(7*a*x**(7/2)) + 2*A*b/(5*a**
2*x**(5/2)) - 2*A*b**2/(3*a**3*x**(3/2)) + A*b**3*log(sqrt(x) - sqrt(-a/b))/(a**4*sqrt(-a/b)) - A*b**3*log(sqr
t(x) + sqrt(-a/b))/(a**4*sqrt(-a/b)) + 2*A*b**3/(a**4*sqrt(x)) - 2*B/(5*a*x**(5/2)) + 2*B*b/(3*a**2*x**(3/2))
- B*b**2*log(sqrt(x) - sqrt(-a/b))/(a**3*sqrt(-a/b)) + B*b**2*log(sqrt(x) + sqrt(-a/b))/(a**3*sqrt(-a/b)) - 2*
B*b**2/(a**3*sqrt(x)), True))

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Giac [A]
time = 1.07, size = 104, normalized size = 0.92 \begin {gather*} -\frac {2 \, {\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {2 \, {\left (105 \, B a b^{2} x^{3} - 105 \, A b^{3} x^{3} - 35 \, B a^{2} b x^{2} + 35 \, A a b^{2} x^{2} + 21 \, B a^{3} x - 21 \, A a^{2} b x + 15 \, A a^{3}\right )}}{105 \, a^{4} x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(B*a*b^3 - A*b^4)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) - 2/105*(105*B*a*b^2*x^3 - 105*A*b^3*x^3 - 35
*B*a^2*b*x^2 + 35*A*a*b^2*x^2 + 21*B*a^3*x - 21*A*a^2*b*x + 15*A*a^3)/(a^4*x^(7/2))

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Mupad [B]
time = 0.42, size = 90, normalized size = 0.80 \begin {gather*} \frac {2\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (A\,b-B\,a\right )}{a^{9/2}}-\frac {\frac {2\,A}{7\,a}-\frac {2\,x\,\left (A\,b-B\,a\right )}{5\,a^2}-\frac {2\,b^2\,x^3\,\left (A\,b-B\,a\right )}{a^4}+\frac {2\,b\,x^2\,\left (A\,b-B\,a\right )}{3\,a^3}}{x^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(9/2)*(a + b*x)),x)

[Out]

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

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