3.696 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=267 \[ -\frac{A (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6}-\frac{a^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{5 a^4 b B \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{10 a^3 b^2 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{5 a b^4 B \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^5 B \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

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

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Rubi [A]  time = 0.0839709, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {770, 78, 43} \[ -\frac{A (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6}-\frac{a^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{5 a^4 b B \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{10 a^3 b^2 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{5 a b^4 B \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^5 B \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^7,x]

[Out]

-(a^5*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^5*(a + b*x)) - (5*a^4*b*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(a
+ b*x)) - (10*a^3*b^2*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (5*a^2*b^3*B*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(x^2*(a + b*x)) - (5*a*b^4*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x*(a + b*x)) - (A*(a + b*x)^5*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(6*a*x^6) + (b^5*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 78

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0453606, size = 127, normalized size = 0.48 \[ -\frac{\sqrt{(a+b x)^2} \left (100 a^2 b^3 x^3 (2 A+3 B x)+50 a^3 b^2 x^2 (3 A+4 B x)+15 a^4 b x (4 A+5 B x)+2 a^5 (5 A+6 B x)+150 a b^4 x^4 (A+2 B x)+60 A b^5 x^5-60 b^5 B x^6 \log (x)\right )}{60 x^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^7,x]

[Out]

-(Sqrt[(a + b*x)^2]*(60*A*b^5*x^5 + 150*a*b^4*x^4*(A + 2*B*x) + 100*a^2*b^3*x^3*(2*A + 3*B*x) + 50*a^3*b^2*x^2
*(3*A + 4*B*x) + 15*a^4*b*x*(4*A + 5*B*x) + 2*a^5*(5*A + 6*B*x) - 60*b^5*B*x^6*Log[x]))/(60*x^6*(a + b*x))

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Maple [A]  time = 0.013, size = 142, normalized size = 0.5 \begin{align*} -{\frac{-60\,B{b}^{5}\ln \left ( x \right ){x}^{6}+60\,A{x}^{5}{b}^{5}+300\,B{x}^{5}a{b}^{4}+150\,A{x}^{4}a{b}^{4}+300\,B{x}^{4}{a}^{2}{b}^{3}+200\,A{x}^{3}{a}^{2}{b}^{3}+200\,B{x}^{3}{a}^{3}{b}^{2}+150\,A{x}^{2}{a}^{3}{b}^{2}+75\,B{x}^{2}{a}^{4}b+60\,A{a}^{4}bx+12\,B{a}^{5}x+10\,A{a}^{5}}{60\, \left ( bx+a \right ) ^{5}{x}^{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((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^7,x)

[Out]

-1/60*((b*x+a)^2)^(5/2)*(-60*B*b^5*ln(x)*x^6+60*A*x^5*b^5+300*B*x^5*a*b^4+150*A*x^4*a*b^4+300*B*x^4*a^2*b^3+20
0*A*x^3*a^2*b^3+200*B*x^3*a^3*b^2+150*A*x^2*a^3*b^2+75*B*x^2*a^4*b+60*A*a^4*b*x+12*B*a^5*x+10*A*a^5)/(b*x+a)^5
/x^6

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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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53504, size = 270, normalized size = 1.01 \begin{align*} \frac{60 \, B b^{5} x^{6} \log \left (x\right ) - 10 \, A a^{5} - 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} - 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \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^7,x, algorithm="fricas")

[Out]

1/60*(60*B*b^5*x^6*log(x) - 10*A*a^5 - 60*(5*B*a*b^4 + A*b^5)*x^5 - 150*(2*B*a^2*b^3 + A*a*b^4)*x^4 - 200*(B*a
^3*b^2 + A*a^2*b^3)*x^3 - 75*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 12*(B*a^5 + 5*A*a^4*b)*x)/x^6

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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^{7}}\, 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**7,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**7, x)

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Giac [A]  time = 1.16146, size = 258, normalized size = 0.97 \begin{align*} B b^{5} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) - \frac{10 \, A a^{5} \mathrm{sgn}\left (b x + a\right ) + 60 \,{\left (5 \, B a b^{4} \mathrm{sgn}\left (b x + a\right ) + A b^{5} \mathrm{sgn}\left (b x + a\right )\right )} x^{5} + 150 \,{\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} + 200 \,{\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} + 75 \,{\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} + 12 \,{\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^{6}} \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^7,x, algorithm="giac")

[Out]

B*b^5*log(abs(x))*sgn(b*x + a) - 1/60*(10*A*a^5*sgn(b*x + a) + 60*(5*B*a*b^4*sgn(b*x + a) + A*b^5*sgn(b*x + a)
)*x^5 + 150*(2*B*a^2*b^3*sgn(b*x + a) + A*a*b^4*sgn(b*x + a))*x^4 + 200*(B*a^3*b^2*sgn(b*x + a) + A*a^2*b^3*sg
n(b*x + a))*x^3 + 75*(B*a^4*b*sgn(b*x + a) + 2*A*a^3*b^2*sgn(b*x + a))*x^2 + 12*(B*a^5*sgn(b*x + a) + 5*A*a^4*
b*sgn(b*x + a))*x)/x^6