3.8.37 \(\int \frac {x^{5/2} (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=238 \[ \frac {2 x^{5/2} (a+b x) (A b-a B)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^2 \sqrt {x} (a+b x) (A b-a B)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a x^{3/2} (a+b x) (A b-a B)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{7/2} (a+b x)}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.11, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {770, 80, 50, 63, 205} \begin {gather*} \frac {2 x^{5/2} (a+b x) (A b-a B)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a x^{3/2} (a+b x) (A b-a B)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^2 \sqrt {x} (a+b x) (A b-a B)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{7/2} (a+b x)}{7 b \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 50

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 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 80

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

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]

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 120, normalized size = 0.50 \begin {gather*} \frac {2 (a+b x) \left (105 a^{5/2} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\sqrt {b} \sqrt {x} \left (-105 a^3 B+35 a^2 b (3 A+B x)-7 a b^2 x (5 A+3 B x)+3 b^3 x^2 (7 A+5 B x)\right )\right )}{105 b^{9/2} \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)*(Sqrt[b]*Sqrt[x]*(-105*a^3*B + 35*a^2*b*(3*A + B*x) - 7*a*b^2*x*(5*A + 3*B*x) + 3*b^3*x^2*(7*A +
5*B*x)) + 105*a^(5/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(105*b^(9/2)*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [A]  time = 11.70, size = 148, normalized size = 0.62 \begin {gather*} \frac {(a+b x) \left (\frac {2 \left (a^{7/2} B-a^{5/2} A b\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2}}+\frac {2 \left (-105 a^3 B \sqrt {x}+105 a^2 A b \sqrt {x}+35 a^2 b B x^{3/2}-35 a A b^2 x^{3/2}-21 a b^2 B x^{5/2}+21 A b^3 x^{5/2}+15 b^3 B x^{7/2}\right )}{105 b^4}\right )}{\sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*((2*(105*a^2*A*b*Sqrt[x] - 105*a^3*B*Sqrt[x] - 35*a*A*b^2*x^(3/2) + 35*a^2*b*B*x^(3/2) + 21*A*b^3*x
^(5/2) - 21*a*b^2*B*x^(5/2) + 15*b^3*B*x^(7/2)))/(105*b^4) + (2*(-(a^(5/2)*A*b) + a^(7/2)*B)*ArcTan[(Sqrt[b]*S
qrt[x])/Sqrt[a]])/b^(9/2)))/Sqrt[(a + b*x)^2]

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fricas [A]  time = 0.47, size = 229, normalized size = 0.96 \begin {gather*} \left [-\frac {105 \, {\left (B a^{3} - A a^{2} b\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 (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt {x}}{105 \, b^{4}}, \frac {2 \, {\left (105 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt {x}\right )}}{105 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.20, size = 169, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (B a^{4} \mathrm {sgn}\left (b x + a\right ) - A a^{3} b \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {2 \, {\left (15 \, B b^{6} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) - 21 \, B a b^{5} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + 21 \, A b^{6} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + 35 \, B a^{2} b^{4} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) - 35 \, A a b^{5} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) - 105 \, B a^{3} b^{3} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) + 105 \, A a^{2} b^{4} \sqrt {x} \mathrm {sgn}\left (b x + a\right )\right )}}{105 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

2*(B*a^4*sgn(b*x + a) - A*a^3*b*sgn(b*x + a))*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) + 2/105*(15*B*b^6*x^
(7/2)*sgn(b*x + a) - 21*B*a*b^5*x^(5/2)*sgn(b*x + a) + 21*A*b^6*x^(5/2)*sgn(b*x + a) + 35*B*a^2*b^4*x^(3/2)*sg
n(b*x + a) - 35*A*a*b^5*x^(3/2)*sgn(b*x + a) - 105*B*a^3*b^3*sqrt(x)*sgn(b*x + a) + 105*A*a^2*b^4*sqrt(x)*sgn(
b*x + a))/b^7

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maple [A]  time = 0.07, size = 163, normalized size = 0.68 \begin {gather*} \frac {2 \left (b x +a \right ) \left (15 \sqrt {a b}\, B \,b^{3} x^{\frac {7}{2}}+21 \sqrt {a b}\, A \,b^{3} x^{\frac {5}{2}}-21 \sqrt {a b}\, B a \,b^{2} x^{\frac {5}{2}}-105 A \,a^{3} b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+105 B \,a^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-35 \sqrt {a b}\, A a \,b^{2} x^{\frac {3}{2}}+35 \sqrt {a b}\, B \,a^{2} b \,x^{\frac {3}{2}}+105 \sqrt {a b}\, A \,a^{2} b \sqrt {x}-105 \sqrt {a b}\, B \,a^{3} \sqrt {x}\right )}{105 \sqrt {\left (b x +a \right )^{2}}\, \sqrt {a b}\, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(B*x+A)/((b*x+a)^2)^(1/2),x)

[Out]

2/105*(b*x+a)*(15*B*x^(7/2)*(a*b)^(1/2)*b^3+21*A*x^(5/2)*(a*b)^(1/2)*b^3-21*B*x^(5/2)*(a*b)^(1/2)*a*b^2-35*A*x
^(3/2)*(a*b)^(1/2)*a*b^2+35*B*x^(3/2)*(a*b)^(1/2)*a^2*b+105*A*x^(1/2)*(a*b)^(1/2)*a^2*b-105*A*arctan(1/(a*b)^(
1/2)*b*x^(1/2))*a^3*b-105*B*x^(1/2)*(a*b)^(1/2)*a^3+105*B*arctan(1/(a*b)^(1/2)*b*x^(1/2))*a^4)/((b*x+a)^2)^(1/
2)/b^4/(a*b)^(1/2)

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maxima [A]  time = 1.74, size = 203, normalized size = 0.85 \begin {gather*} \frac {6 \, {\left (5 \, B b^{3} x^{2} + 7 \, B a b^{2} x\right )} x^{\frac {5}{2}} - 2 \, {\left (3 \, {\left (9 \, B a b^{2} - 7 \, A b^{3}\right )} x^{2} + 7 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} x^{\frac {3}{2}} - 7 \, {\left (3 \, {\left (9 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 5 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {x}}{105 \, {\left (b^{4} x + a b^{3}\right )}} + \frac {2 \, {\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {{\left (9 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{\frac {3}{2}} - 6 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {x}}{3 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^{5/2}\,\left (A+B\,x\right )}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(5/2)*(A + B*x))/((a + b*x)^2)^(1/2),x)

[Out]

int((x^(5/2)*(A + B*x))/((a + b*x)^2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x+A)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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