3.695 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=293 \[ -\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{4 x^4 (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{3 x^3 (a+b x)}-\frac{5 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x^2 (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{x (a+b x)}+\frac{b^4 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{a+b x}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}+\frac{b^5 B x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

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

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Rubi [A]  time = 0.116138, antiderivative size = 293, 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, 76} \[ -\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{4 x^4 (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{3 x^3 (a+b x)}-\frac{5 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x^2 (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{x (a+b x)}+\frac{b^4 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{a+b x}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}+\frac{b^5 B x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^6,x]

[Out]

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

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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^6} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5 (A+B x)}{x^6} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (b^{10} B+\frac{a^5 A b^5}{x^6}+\frac{a^4 b^5 (5 A b+a B)}{x^5}+\frac{5 a^3 b^6 (2 A b+a B)}{x^4}+\frac{10 a^2 b^7 (A b+a B)}{x^3}+\frac{5 a b^8 (A b+2 a B)}{x^2}+\frac{b^9 (A b+5 a B)}{x}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{a^4 (5 A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{5 a^3 b (2 A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^2 b^2 (A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{5 a b^3 (A b+2 a B) \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^5 B x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{b^4 (A b+5 a B) \sqrt{a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end{align*}

Mathematica [A]  time = 0.0545261, size = 127, normalized size = 0.43 \[ -\frac{\sqrt{(a+b x)^2} \left (300 a^2 b^3 x^3 (A+2 B x)+100 a^3 b^2 x^2 (2 A+3 B x)+25 a^4 b x (3 A+4 B x)+3 a^5 (4 A+5 B x)-60 b^4 x^5 \log (x) (5 a B+A b)+300 a A b^4 x^4-60 b^5 B x^6\right )}{60 x^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(300*a*A*b^4*x^4 - 60*b^5*B*x^6 + 300*a^2*b^3*x^3*(A + 2*B*x) + 100*a^3*b^2*x^2*(2*A + 3*B
*x) + 25*a^4*b*x*(3*A + 4*B*x) + 3*a^5*(4*A + 5*B*x) - 60*b^4*(A*b + 5*a*B)*x^5*Log[x]))/(60*x^5*(a + b*x))

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Maple [A]  time = 0.014, size = 144, normalized size = 0.5 \begin{align*}{\frac{60\,A\ln \left ( x \right ){x}^{5}{b}^{5}+300\,B\ln \left ( x \right ){x}^{5}a{b}^{4}+60\,B{b}^{5}{x}^{6}-300\,A{x}^{4}a{b}^{4}-600\,B{x}^{4}{a}^{2}{b}^{3}-300\,A{x}^{3}{a}^{2}{b}^{3}-300\,B{x}^{3}{a}^{3}{b}^{2}-200\,A{x}^{2}{a}^{3}{b}^{2}-100\,B{x}^{2}{a}^{4}b-75\,A{a}^{4}bx-15\,B{a}^{5}x-12\,A{a}^{5}}{60\, \left ( bx+a \right ) ^{5}{x}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^6,x)

[Out]

1/60*((b*x+a)^2)^(5/2)*(60*A*ln(x)*x^5*b^5+300*B*ln(x)*x^5*a*b^4+60*B*b^5*x^6-300*A*x^4*a*b^4-600*B*x^4*a^2*b^
3-300*A*x^3*a^2*b^3-300*B*x^3*a^3*b^2-200*A*x^2*a^3*b^2-100*B*x^2*a^4*b-75*A*a^4*b*x-15*B*a^5*x-12*A*a^5)/(b*x
+a)^5/x^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.36671, size = 271, normalized size = 0.92 \begin{align*} \frac{60 \, B b^{5} x^{6} + 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} \log \left (x\right ) - 12 \, A a^{5} - 300 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 100 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 15 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^6,x, algorithm="fricas")

[Out]

1/60*(60*B*b^5*x^6 + 60*(5*B*a*b^4 + A*b^5)*x^5*log(x) - 12*A*a^5 - 300*(2*B*a^2*b^3 + A*a*b^4)*x^4 - 300*(B*a
^3*b^2 + A*a^2*b^3)*x^3 - 100*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 15*(B*a^5 + 5*A*a^4*b)*x)/x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/x**6,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**6, x)

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Giac [A]  time = 1.16399, size = 254, normalized size = 0.87 \begin{align*} B b^{5} x \mathrm{sgn}\left (b x + a\right ) +{\left (5 \, B a b^{4} \mathrm{sgn}\left (b x + a\right ) + A b^{5} \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right ) - \frac{12 \, A a^{5} \mathrm{sgn}\left (b x + a\right ) + 300 \,{\left (2 \, B a^{2} b^{3} \mathrm{sgn}\left (b x + a\right ) + A a b^{4} \mathrm{sgn}\left (b x + a\right )\right )} x^{4} + 300 \,{\left (B a^{3} b^{2} \mathrm{sgn}\left (b x + a\right ) + A a^{2} b^{3} \mathrm{sgn}\left (b x + a\right )\right )} x^{3} + 100 \,{\left (B a^{4} b \mathrm{sgn}\left (b x + a\right ) + 2 \, A a^{3} b^{2} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 15 \,{\left (B a^{5} \mathrm{sgn}\left (b x + a\right ) + 5 \, A a^{4} b \mathrm{sgn}\left (b x + a\right )\right )} x}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^6,x, algorithm="giac")

[Out]

B*b^5*x*sgn(b*x + a) + (5*B*a*b^4*sgn(b*x + a) + A*b^5*sgn(b*x + a))*log(abs(x)) - 1/60*(12*A*a^5*sgn(b*x + a)
+ 300*(2*B*a^2*b^3*sgn(b*x + a) + A*a*b^4*sgn(b*x + a))*x^4 + 300*(B*a^3*b^2*sgn(b*x + a) + A*a^2*b^3*sgn(b*x
+ a))*x^3 + 100*(B*a^4*b*sgn(b*x + a) + 2*A*a^3*b^2*sgn(b*x + a))*x^2 + 15*(B*a^5*sgn(b*x + a) + 5*A*a^4*b*sg
n(b*x + a))*x)/x^5