3.714 \(\int \frac{x^4 (A+B x)}{(a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{a^4 (A b-a B)}{2 b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^3 (4 A b-5 a B)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a x (a+b x) (A b-2 a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x) (A b-3 a B)}{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 (a+b x) (3 A b-5 a B) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^3 (a+b x)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(a^3*(4*A*b - 5*a*B))/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a^4*(A*b - a*B))/(2*b^6*(a + b*x)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2]) - (3*a*(A*b - 2*a*B)*x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - 3*a*B)*x^2*(a
+ b*x))/(2*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (B*x^3*(a + b*x))/(3*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*a
^2*(3*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 = 0.180792, antiderivative size = 249, 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} \[ -\frac{a^4 (A b-a B)}{2 b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^3 (4 A b-5 a B)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a x (a+b x) (A b-2 a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x) (A b-3 a B)}{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 (a+b x) (3 A b-5 a B) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^3 (a+b x)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(a^3*(4*A*b - 5*a*B))/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a^4*(A*b - a*B))/(2*b^6*(a + b*x)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2]) - (3*a*(A*b - 2*a*B)*x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - 3*a*B)*x^2*(a
+ b*x))/(2*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (B*x^3*(a + b*x))/(3*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*a
^2*(3*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 )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{x^4 (A+B x)}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac{3 a (-A b+2 a B)}{b^8}+\frac{(A b-3 a B) x}{b^7}+\frac{B x^2}{b^6}-\frac{a^4 (-A b+a B)}{b^8 (a+b x)^3}+\frac{a^3 (-4 A b+5 a B)}{b^8 (a+b x)^2}-\frac{2 a^2 (-3 A b+5 a B)}{b^8 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{a^3 (4 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)}{2 b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a (A b-2 a B) x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-3 a B) x^2 (a+b x)}{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^3 (a+b x)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 (3 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 = 0.0674365, size = 140, normalized size = 0.56 \[ \frac{a^2 b^3 x^2 (20 B x-33 A)+3 a^3 b^2 x (2 A+21 B x)+3 a^4 b (7 A+2 B x)-12 a^2 (a+b x)^2 (5 a B-3 A b) \log (a+b x)-27 a^5 B-a b^4 x^3 (12 A+5 B x)+b^5 x^4 (3 A+2 B x)}{6 b^6 (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(-27*a^5*B + b^5*x^4*(3*A + 2*B*x) + 3*a^4*b*(7*A + 2*B*x) - a*b^4*x^3*(12*A + 5*B*x) + a^2*b^3*x^2*(-33*A + 2
0*B*x) + 3*a^3*b^2*x*(2*A + 21*B*x) - 12*a^2*(-3*A*b + 5*a*B)*(a + b*x)^2*Log[a + b*x])/(6*b^6*(a + b*x)*Sqrt[
(a + b*x)^2])

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Maple [A]  time = 0.017, size = 217, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,B{b}^{5}{x}^{5}+3\,A{x}^{4}{b}^{5}-5\,B{x}^{4}a{b}^{4}+36\,A\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}-12\,A{x}^{3}a{b}^{4}-60\,B\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}+20\,B{x}^{3}{a}^{2}{b}^{3}+72\,A\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}-33\,A{x}^{2}{a}^{2}{b}^{3}-120\,B\ln \left ( bx+a \right ) x{a}^{4}b+63\,B{x}^{2}{a}^{3}{b}^{2}+36\,A\ln \left ( bx+a \right ){a}^{4}b+6\,A{a}^{3}{b}^{2}x-60\,B\ln \left ( bx+a \right ){a}^{5}+6\,B{a}^{4}bx+21\,A{a}^{4}b-27\,B{a}^{5} \right ) \left ( bx+a \right ) }{6\,{b}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{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)^(3/2),x)

[Out]

1/6*(2*B*b^5*x^5+3*A*x^4*b^5-5*B*x^4*a*b^4+36*A*ln(b*x+a)*x^2*a^2*b^3-12*A*x^3*a*b^4-60*B*ln(b*x+a)*x^2*a^3*b^
2+20*B*x^3*a^2*b^3+72*A*ln(b*x+a)*x*a^3*b^2-33*A*x^2*a^2*b^3-120*B*ln(b*x+a)*x*a^4*b+63*B*x^2*a^3*b^2+36*A*ln(
b*x+a)*a^4*b+6*A*a^3*b^2*x-60*B*ln(b*x+a)*a^5+6*B*a^4*b*x+21*A*a^4*b-27*B*a^5)*(b*x+a)/b^6/((b*x+a)^2)^(3/2)

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Maxima [B]  time = 0.97411, size = 508, normalized size = 2.04 \begin{align*} \frac{B x^{4}}{3 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{7 \, B a x^{3}}{6 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac{A x^{3}}{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{9 \, B a^{2} x^{2}}{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac{5 \, A a x^{2}}{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} - \frac{10 \, B a^{3} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{3}} + \frac{6 \, A a^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{2}} + \frac{9 \, A a^{4}}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{15 \, B a^{5}}{{\left (b^{2}\right )}^{\frac{7}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} - \frac{20 \, B a^{4} x}{{\left (b^{2}\right )}^{\frac{5}{2}} b^{2}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{12 \, A a^{3} x}{{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} + \frac{9 \, B a^{4}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{6}} - \frac{5 \, A a^{3}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} - \frac{9 \, B a^{5}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b^{5}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{5 \, A a^{4}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b^{4}{\left (x + \frac{a}{b}\right )}^{2}} \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)^(3/2),x, algorithm="maxima")

[Out]

1/3*B*x^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 7/6*B*a*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^3) + 1/2*A*x^3/(s
qrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) + 9/2*B*a^2*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^4) - 5/2*A*a*x^2/(sqrt(b^2*
x^2 + 2*a*b*x + a^2)*b^3) - 10*B*a^3*log(x + a/b)/((b^2)^(3/2)*b^3) + 6*A*a^2*log(x + a/b)/((b^2)^(3/2)*b^2) +
9*A*a^4/((b^2)^(7/2)*(x + a/b)^2) - 15*B*a^5/((b^2)^(7/2)*b*(x + a/b)^2) - 20*B*a^4*x/((b^2)^(5/2)*b^2*(x + a
/b)^2) + 12*A*a^3*x/((b^2)^(5/2)*b*(x + a/b)^2) + 9*B*a^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^6) - 5*A*a^3/(sqrt(
b^2*x^2 + 2*a*b*x + a^2)*b^5) - 9/2*B*a^5/((b^2)^(3/2)*b^5*(x + a/b)^2) + 5/2*A*a^4/((b^2)^(3/2)*b^4*(x + a/b)
^2)

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Fricas [A]  time = 1.66129, size = 417, normalized size = 1.67 \begin{align*} \frac{2 \, B b^{5} x^{5} - 27 \, B a^{5} + 21 \, A a^{4} b -{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \,{\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 3 \,{\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} + 6 \,{\left (B a^{4} b + A a^{3} b^{2}\right )} x - 12 \,{\left (5 \, B a^{5} - 3 \, A a^{4} b +{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} 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)^(3/2),x, algorithm="fricas")

[Out]

1/6*(2*B*b^5*x^5 - 27*B*a^5 + 21*A*a^4*b - (5*B*a*b^4 - 3*A*b^5)*x^4 + 4*(5*B*a^2*b^3 - 3*A*a*b^4)*x^3 + 3*(21
*B*a^3*b^2 - 11*A*a^2*b^3)*x^2 + 6*(B*a^4*b + A*a^3*b^2)*x - 12*(5*B*a^5 - 3*A*a^4*b + (5*B*a^3*b^2 - 3*A*a^2*
b^3)*x^2 + 2*(5*B*a^4*b - 3*A*a^3*b^2)*x)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x + a^2*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{3}{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)**(3/2),x)

[Out]

Integral(x**4*(A + B*x)/((a + b*x)**2)**(3/2), 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)^(3/2),x, algorithm="giac")

[Out]

sage0*x