3.722 \(\int \frac{x^4 (A+B x)}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=245 \[ -\frac{a^4 (A b-a B)}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^3 (4 A b-5 a B)}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^2 (3 A b-5 a B)}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a (2 A b-5 a B)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (A b-5 a B) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(2*a*(2*A*b - 5*a*B))/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a^4*(A*b - a*B))/(4*b^6*(a + b*x)^3*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) + (a^3*(4*A*b - 5*a*B))/(3*b^6*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a^2*(3*A*b - 5*
a*B))/(b^6*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (B*x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((
A*b - 5*a*B)*(a + b*x)*Log[a + b*x])/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.179413, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {770, 77} \[ -\frac{a^4 (A b-a B)}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^3 (4 A b-5 a B)}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^2 (3 A b-5 a B)}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a (2 A b-5 a B)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (A b-5 a B) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*a*(2*A*b - 5*a*B))/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a^4*(A*b - a*B))/(4*b^6*(a + b*x)^3*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) + (a^3*(4*A*b - 5*a*B))/(3*b^6*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a^2*(3*A*b - 5*
a*B))/(b^6*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (B*x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((
A*b - 5*a*B)*(a + b*x)*Log[a + b*x])/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{x^4 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{x^4 (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac{B}{b^{10}}-\frac{a^4 (-A b+a B)}{b^{10} (a+b x)^5}+\frac{a^3 (-4 A b+5 a B)}{b^{10} (a+b x)^4}-\frac{2 a^2 (-3 A b+5 a B)}{b^{10} (a+b x)^3}+\frac{2 a (-2 A b+5 a B)}{b^{10} (a+b x)^2}+\frac{A b-5 a B}{b^{10} (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 a (2 A b-5 a B)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^4 (A b-a B)}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^3 (4 A b-5 a B)}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^2 (3 A b-5 a B)}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-5 a B) (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0719678, size = 127, normalized size = 0.52 \[ \frac{12 a^2 b^3 x^2 (9 A-4 B x)+4 a^3 b^2 x (22 A-63 B x)+a^4 b (25 A-248 B x)-77 a^5 B+48 a b^4 x^3 (A+B x)+12 (a+b x)^4 (A b-5 a B) \log (a+b x)+12 b^5 B x^5}{12 b^6 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-77*a^5*B + 12*b^5*B*x^5 + a^4*b*(25*A - 248*B*x) + 4*a^3*b^2*x*(22*A - 63*B*x) + 12*a^2*b^3*x^2*(9*A - 4*B*x
) + 48*a*b^4*x^3*(A + B*x) + 12*(A*b - 5*a*B)*(a + b*x)^4*Log[a + b*x])/(12*b^6*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.013, size = 273, normalized size = 1.1 \begin{align*}{\frac{ \left ( 12\,A\ln \left ( bx+a \right ){x}^{4}{b}^{5}-60\,B\ln \left ( bx+a \right ){x}^{4}a{b}^{4}+12\,B{b}^{5}{x}^{5}+48\,A\ln \left ( bx+a \right ){x}^{3}a{b}^{4}-240\,B\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{3}+48\,B{x}^{4}a{b}^{4}+72\,A\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}+48\,A{x}^{3}a{b}^{4}-360\,B\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}-48\,B{x}^{3}{a}^{2}{b}^{3}+48\,A\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}+108\,A{x}^{2}{a}^{2}{b}^{3}-240\,B\ln \left ( bx+a \right ) x{a}^{4}b-252\,B{x}^{2}{a}^{3}{b}^{2}+12\,A\ln \left ( bx+a \right ){a}^{4}b+88\,A{a}^{3}{b}^{2}x-60\,B\ln \left ( bx+a \right ){a}^{5}-248\,B{a}^{4}bx+25\,A{a}^{4}b-77\,B{a}^{5} \right ) \left ( bx+a \right ) }{12\,{b}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/12*(12*A*ln(b*x+a)*x^4*b^5-60*B*ln(b*x+a)*x^4*a*b^4+12*B*b^5*x^5+48*A*ln(b*x+a)*x^3*a*b^4-240*B*ln(b*x+a)*x^
3*a^2*b^3+48*B*x^4*a*b^4+72*A*ln(b*x+a)*x^2*a^2*b^3+48*A*x^3*a*b^4-360*B*ln(b*x+a)*x^2*a^3*b^2-48*B*x^3*a^2*b^
3+48*A*ln(b*x+a)*x*a^3*b^2+108*A*x^2*a^2*b^3-240*B*ln(b*x+a)*x*a^4*b-252*B*x^2*a^3*b^2+12*A*ln(b*x+a)*a^4*b+88
*A*a^3*b^2*x-60*B*ln(b*x+a)*a^5-248*B*a^4*b*x+25*A*a^4*b-77*B*a^5)*(b*x+a)/b^6/((b*x+a)^2)^(5/2)

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Maxima [A]  time = 1.20558, size = 285, normalized size = 1.16 \begin{align*} \frac{1}{12} \, B{\left (\frac{12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}} - \frac{60 \, a \log \left (b x + a\right )}{b^{6}}\right )} + \frac{1}{12} \, A{\left (\frac{48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}} + \frac{12 \, \log \left (b x + a\right )}{b^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/12*B*((12*b^5*x^5 + 48*a*b^4*x^4 - 48*a^2*b^3*x^3 - 252*a^3*b^2*x^2 - 248*a^4*b*x - 77*a^5)/(b^10*x^4 + 4*a*
b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6) - 60*a*log(b*x + a)/b^6) + 1/12*A*((48*a*b^3*x^3 + 108*a^2*b^
2*x^2 + 88*a^3*b*x + 25*a^4)/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5) + 12*log(b*x + a)
/b^5)

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Fricas [A]  time = 1.818, size = 525, normalized size = 2.14 \begin{align*} \frac{12 \, B b^{5} x^{5} + 48 \, B a b^{4} x^{4} - 77 \, B a^{5} + 25 \, A a^{4} b - 48 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} x^{3} - 36 \,{\left (7 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} - 8 \,{\left (31 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x - 12 \,{\left (5 \, B a^{5} - A a^{4} b +{\left (5 \, B a b^{4} - A b^{5}\right )} x^{4} + 4 \,{\left (5 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 6 \,{\left (5 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (5 \, B a^{4} b - A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12*(12*B*b^5*x^5 + 48*B*a*b^4*x^4 - 77*B*a^5 + 25*A*a^4*b - 48*(B*a^2*b^3 - A*a*b^4)*x^3 - 36*(7*B*a^3*b^2 -
3*A*a^2*b^3)*x^2 - 8*(31*B*a^4*b - 11*A*a^3*b^2)*x - 12*(5*B*a^5 - A*a^4*b + (5*B*a*b^4 - A*b^5)*x^4 + 4*(5*B
*a^2*b^3 - A*a*b^4)*x^3 + 6*(5*B*a^3*b^2 - A*a^2*b^3)*x^2 + 4*(5*B*a^4*b - A*a^3*b^2)*x)*log(b*x + a))/(b^10*x
^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(x**4*(A + B*x)/((a + b*x)**2)**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x