3.704 \(\int \frac{x^4 (A+B x)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=258 \[ \frac{x^4 (a+b x) (A b-a B)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a x^3 (a+b x) (A b-a B)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 x^2 (a+b x) (A b-a B)}{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^3 x (a+b x) (A b-a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^4 (a+b x) (A b-a B) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^5 (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

-((a^3*(A*b - a*B)*x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + (a^2*(A*b - a*B)*x^2*(a + b*x))/(2*b^4*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a*(A*b - a*B)*x^3*(a + b*x))/(3*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b -
a*B)*x^4*(a + b*x))/(4*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (B*x^5*(a + b*x))/(5*b*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) + (a^4*(A*b - 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 = 0.153521, antiderivative size = 258, 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} \[ \frac{x^4 (a+b x) (A b-a B)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a x^3 (a+b x) (A b-a B)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 x^2 (a+b x) (A b-a B)}{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^3 x (a+b x) (A b-a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^4 (a+b x) (A b-a B) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^5 (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^3*(A*b - a*B)*x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + (a^2*(A*b - a*B)*x^2*(a + b*x))/(2*b^4*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a*(A*b - a*B)*x^3*(a + b*x))/(3*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b -
a*B)*x^4*(a + b*x))/(4*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (B*x^5*(a + b*x))/(5*b*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) + (a^4*(A*b - 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)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{x^4 (A+B x)}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{a^3 (-A b+a B)}{b^6}-\frac{a^2 (-A b+a B) x}{b^5}+\frac{a (-A b+a B) x^2}{b^4}+\frac{(A b-a B) x^3}{b^3}+\frac{B x^4}{b^2}-\frac{a^4 (-A b+a B)}{b^6 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{a^3 (A b-a B) x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 (A b-a B) x^2 (a+b x)}{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a (A b-a B) x^3 (a+b x)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^4 (a+b x)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^5 (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^4 (A b-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 = 0.0615225, size = 116, normalized size = 0.45 \[ \frac{(a+b x) \left (b x \left (10 a^2 b^2 x (3 A+2 B x)-30 a^3 b (2 A+B x)+60 a^4 B-5 a b^3 x^2 (4 A+3 B x)+3 b^4 x^3 (5 A+4 B x)\right )-60 a^4 (a B-A b) \log (a+b x)\right )}{60 b^6 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*(b*x*(60*a^4*B - 30*a^3*b*(2*A + B*x) + 10*a^2*b^2*x*(3*A + 2*B*x) - 5*a*b^3*x^2*(4*A + 3*B*x) + 3*
b^4*x^3*(5*A + 4*B*x)) - 60*a^4*(-(A*b) + a*B)*Log[a + b*x]))/(60*b^6*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.007, size = 138, normalized size = 0.5 \begin{align*}{\frac{ \left ( bx+a \right ) \left ( 12\,B{b}^{5}{x}^{5}+15\,A{x}^{4}{b}^{5}-15\,B{x}^{4}a{b}^{4}-20\,A{x}^{3}a{b}^{4}+20\,B{x}^{3}{a}^{2}{b}^{3}+30\,A{x}^{2}{a}^{2}{b}^{3}-30\,B{x}^{2}{a}^{3}{b}^{2}+60\,A\ln \left ( bx+a \right ){a}^{4}b-60\,A{a}^{3}{b}^{2}x-60\,B\ln \left ( bx+a \right ){a}^{5}+60\,B{a}^{4}bx \right ) }{60\,{b}^{6}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)/((b*x+a)^2)^(1/2),x)

[Out]

1/60*(b*x+a)*(12*B*b^5*x^5+15*A*x^4*b^5-15*B*x^4*a*b^4-20*A*x^3*a*b^4+20*B*x^3*a^2*b^3+30*A*x^2*a^2*b^3-30*B*x
^2*a^3*b^2+60*A*ln(b*x+a)*a^4*b-60*A*a^3*b^2*x-60*B*ln(b*x+a)*a^5+60*B*a^4*b*x)/((b*x+a)^2)^(1/2)/b^6

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Maxima [A]  time = 1.04986, size = 464, normalized size = 1.8 \begin{align*} \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B x^{4}}{5 \, b^{2}} + \frac{13 \, A a^{4} \log \left (x + \frac{a}{b}\right )}{6 \,{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{77 \, B a^{5} \log \left (x + \frac{a}{b}\right )}{30 \,{\left (b^{2}\right )}^{\frac{5}{2}} b} + \frac{77 \, B a^{4} x}{30 \,{\left (b^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{13 \, A a^{3} x}{6 \,{\left (b^{2}\right )}^{\frac{3}{2}} b} - \frac{77 \, B a^{3} x^{2}}{60 \, \sqrt{b^{2}} b^{3}} + \frac{13 \, A a^{2} x^{2}}{12 \, \sqrt{b^{2}} b^{2}} - \frac{9 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B a x^{3}}{20 \, b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} A x^{3}}{4 \, b^{2}} + \frac{47 \, B a^{5} \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{30 \, b^{5}} - \frac{7 \, A a^{4} \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{6 \, b^{4}} + \frac{47 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{2} x^{2}}{60 \, b^{4}} - \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} A a x^{2}}{12 \, b^{3}} - \frac{47 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{4}}{30 \, b^{6}} + \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} A a^{3}}{6 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/5*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*x^4/b^2 + 13/6*A*a^4*log(x + a/b)/(b^2)^(5/2) - 77/30*B*a^5*log(x + a/b)/(
(b^2)^(5/2)*b) + 77/30*B*a^4*x/((b^2)^(3/2)*b^2) - 13/6*A*a^3*x/((b^2)^(3/2)*b) - 77/60*B*a^3*x^2/(sqrt(b^2)*b
^3) + 13/12*A*a^2*x^2/(sqrt(b^2)*b^2) - 9/20*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a*x^3/b^3 + 1/4*sqrt(b^2*x^2 + 2*
a*b*x + a^2)*A*x^3/b^2 + 47/30*B*a^5*sqrt(b^(-2))*log(x + a/b)/b^5 - 7/6*A*a^4*sqrt(b^(-2))*log(x + a/b)/b^4 +
47/60*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^2*x^2/b^4 - 7/12*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*a*x^2/b^3 - 47/30*sq
rt(b^2*x^2 + 2*a*b*x + a^2)*B*a^4/b^6 + 7/6*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*a^3/b^5

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Fricas [A]  time = 1.65504, size = 244, normalized size = 0.95 \begin{align*} \frac{12 \, B b^{5} x^{5} - 15 \,{\left (B a b^{4} - A b^{5}\right )} x^{4} + 20 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} x^{3} - 30 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 60 \,{\left (B a^{4} b - A a^{3} b^{2}\right )} x - 60 \,{\left (B a^{5} - A a^{4} b\right )} \log \left (b x + a\right )}{60 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/60*(12*B*b^5*x^5 - 15*(B*a*b^4 - A*b^5)*x^4 + 20*(B*a^2*b^3 - A*a*b^4)*x^3 - 30*(B*a^3*b^2 - A*a^2*b^3)*x^2
+ 60*(B*a^4*b - A*a^3*b^2)*x - 60*(B*a^5 - A*a^4*b)*log(b*x + a))/b^6

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Sympy [A]  time = 0.46025, size = 99, normalized size = 0.38 \begin{align*} \frac{B x^{5}}{5 b} - \frac{a^{4} \left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{6}} - \frac{x^{4} \left (- A b + B a\right )}{4 b^{2}} + \frac{x^{3} \left (- A a b + B a^{2}\right )}{3 b^{3}} - \frac{x^{2} \left (- A a^{2} b + B a^{3}\right )}{2 b^{4}} + \frac{x \left (- A a^{3} b + B a^{4}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)/((b*x+a)**2)**(1/2),x)

[Out]

B*x**5/(5*b) - a**4*(-A*b + B*a)*log(a + b*x)/b**6 - x**4*(-A*b + B*a)/(4*b**2) + x**3*(-A*a*b + B*a**2)/(3*b*
*3) - x**2*(-A*a**2*b + B*a**3)/(2*b**4) + x*(-A*a**3*b + B*a**4)/b**5

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Giac [A]  time = 1.39122, size = 250, normalized size = 0.97 \begin{align*} \frac{12 \, B b^{4} x^{5} \mathrm{sgn}\left (b x + a\right ) - 15 \, B a b^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + 15 \, A b^{4} x^{4} \mathrm{sgn}\left (b x + a\right ) + 20 \, B a^{2} b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) - 20 \, A a b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) - 30 \, B a^{3} b x^{2} \mathrm{sgn}\left (b x + a\right ) + 30 \, A a^{2} b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 60 \, B a^{4} x \mathrm{sgn}\left (b x + a\right ) - 60 \, A a^{3} b x \mathrm{sgn}\left (b x + a\right )}{60 \, b^{5}} - \frac{{\left (B a^{5} \mathrm{sgn}\left (b x + a\right ) - A a^{4} b \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/60*(12*B*b^4*x^5*sgn(b*x + a) - 15*B*a*b^3*x^4*sgn(b*x + a) + 15*A*b^4*x^4*sgn(b*x + a) + 20*B*a^2*b^2*x^3*s
gn(b*x + a) - 20*A*a*b^3*x^3*sgn(b*x + a) - 30*B*a^3*b*x^2*sgn(b*x + a) + 30*A*a^2*b^2*x^2*sgn(b*x + a) + 60*B
*a^4*x*sgn(b*x + a) - 60*A*a^3*b*x*sgn(b*x + a))/b^5 - (B*a^5*sgn(b*x + a) - A*a^4*b*sgn(b*x + a))*log(abs(b*x
+ a))/b^6