3.433 \(\int \frac{A+B x}{x^6 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=177 \[ \frac{7 b^2 \sqrt{a+b x} (9 A b-10 a B)}{192 a^4 x^2}-\frac{7 b^3 \sqrt{a+b x} (9 A b-10 a B)}{128 a^5 x}+\frac{7 b^4 (9 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{11/2}}-\frac{7 b \sqrt{a+b x} (9 A b-10 a B)}{240 a^3 x^3}+\frac{\sqrt{a+b x} (9 A b-10 a B)}{40 a^2 x^4}-\frac{A \sqrt{a+b x}}{5 a x^5} \]

[Out]

-(A*Sqrt[a + b*x])/(5*a*x^5) + ((9*A*b - 10*a*B)*Sqrt[a + b*x])/(40*a^2*x^4) - (7*b*(9*A*b - 10*a*B)*Sqrt[a +
b*x])/(240*a^3*x^3) + (7*b^2*(9*A*b - 10*a*B)*Sqrt[a + b*x])/(192*a^4*x^2) - (7*b^3*(9*A*b - 10*a*B)*Sqrt[a +
b*x])/(128*a^5*x) + (7*b^4*(9*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(11/2))

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Rubi [A]  time = 0.079705, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 51, 63, 208} \[ \frac{7 b^2 \sqrt{a+b x} (9 A b-10 a B)}{192 a^4 x^2}-\frac{7 b^3 \sqrt{a+b x} (9 A b-10 a B)}{128 a^5 x}+\frac{7 b^4 (9 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{11/2}}-\frac{7 b \sqrt{a+b x} (9 A b-10 a B)}{240 a^3 x^3}+\frac{\sqrt{a+b x} (9 A b-10 a B)}{40 a^2 x^4}-\frac{A \sqrt{a+b x}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^6*Sqrt[a + b*x]),x]

[Out]

-(A*Sqrt[a + b*x])/(5*a*x^5) + ((9*A*b - 10*a*B)*Sqrt[a + b*x])/(40*a^2*x^4) - (7*b*(9*A*b - 10*a*B)*Sqrt[a +
b*x])/(240*a^3*x^3) + (7*b^2*(9*A*b - 10*a*B)*Sqrt[a + b*x])/(192*a^4*x^2) - (7*b^3*(9*A*b - 10*a*B)*Sqrt[a +
b*x])/(128*a^5*x) + (7*b^4*(9*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(11/2))

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 51

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{A+B x}{x^6 \sqrt{a+b x}} \, dx &=-\frac{A \sqrt{a+b x}}{5 a x^5}+\frac{\left (-\frac{9 A b}{2}+5 a B\right ) \int \frac{1}{x^5 \sqrt{a+b x}} \, dx}{5 a}\\ &=-\frac{A \sqrt{a+b x}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x}}{40 a^2 x^4}+\frac{(7 b (9 A b-10 a B)) \int \frac{1}{x^4 \sqrt{a+b x}} \, dx}{80 a^2}\\ &=-\frac{A \sqrt{a+b x}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x}}{40 a^2 x^4}-\frac{7 b (9 A b-10 a B) \sqrt{a+b x}}{240 a^3 x^3}-\frac{\left (7 b^2 (9 A b-10 a B)\right ) \int \frac{1}{x^3 \sqrt{a+b x}} \, dx}{96 a^3}\\ &=-\frac{A \sqrt{a+b x}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x}}{40 a^2 x^4}-\frac{7 b (9 A b-10 a B) \sqrt{a+b x}}{240 a^3 x^3}+\frac{7 b^2 (9 A b-10 a B) \sqrt{a+b x}}{192 a^4 x^2}+\frac{\left (7 b^3 (9 A b-10 a B)\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{128 a^4}\\ &=-\frac{A \sqrt{a+b x}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x}}{40 a^2 x^4}-\frac{7 b (9 A b-10 a B) \sqrt{a+b x}}{240 a^3 x^3}+\frac{7 b^2 (9 A b-10 a B) \sqrt{a+b x}}{192 a^4 x^2}-\frac{7 b^3 (9 A b-10 a B) \sqrt{a+b x}}{128 a^5 x}-\frac{\left (7 b^4 (9 A b-10 a B)\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{256 a^5}\\ &=-\frac{A \sqrt{a+b x}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x}}{40 a^2 x^4}-\frac{7 b (9 A b-10 a B) \sqrt{a+b x}}{240 a^3 x^3}+\frac{7 b^2 (9 A b-10 a B) \sqrt{a+b x}}{192 a^4 x^2}-\frac{7 b^3 (9 A b-10 a B) \sqrt{a+b x}}{128 a^5 x}-\frac{\left (7 b^3 (9 A b-10 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{128 a^5}\\ &=-\frac{A \sqrt{a+b x}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x}}{40 a^2 x^4}-\frac{7 b (9 A b-10 a B) \sqrt{a+b x}}{240 a^3 x^3}+\frac{7 b^2 (9 A b-10 a B) \sqrt{a+b x}}{192 a^4 x^2}-\frac{7 b^3 (9 A b-10 a B) \sqrt{a+b x}}{128 a^5 x}+\frac{7 b^4 (9 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.0160645, size = 57, normalized size = 0.32 \[ -\frac{\sqrt{a+b x} \left (a^5 A+b^4 x^5 (10 a B-9 A b) \, _2F_1\left (\frac{1}{2},5;\frac{3}{2};\frac{b x}{a}+1\right )\right )}{5 a^6 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + b*x]*(a^5*A + b^4*(-9*A*b + 10*a*B)*x^5*Hypergeometric2F1[1/2, 5, 3/2, 1 + (b*x)/a]))/(5*a^6*x^5)

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Maple [A]  time = 0.011, size = 146, normalized size = 0.8 \begin{align*} 2\,{b}^{4} \left ({\frac{1}{{b}^{5}{x}^{5}} \left ( -{\frac{ \left ( 63\,Ab-70\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{256\,{a}^{5}}}+{\frac{ \left ( 441\,Ab-490\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{384\,{a}^{4}}}-{\frac{ \left ( 63\,Ab-70\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{30\,{a}^{3}}}+{\frac{ \left ( 711\,Ab-790\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{384\,{a}^{2}}}-{\frac{ \left ( 193\,Ab-186\,Ba \right ) \sqrt{bx+a}}{256\,a}} \right ) }+{\frac{63\,Ab-70\,Ba}{256\,{a}^{11/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^6/(b*x+a)^(1/2),x)

[Out]

2*b^4*((-7/256*(9*A*b-10*B*a)/a^5*(b*x+a)^(9/2)+49/384/a^4*(9*A*b-10*B*a)*(b*x+a)^(7/2)-7/30/a^3*(9*A*b-10*B*a
)*(b*x+a)^(5/2)+79/384/a^2*(9*A*b-10*B*a)*(b*x+a)^(3/2)-1/256*(193*A*b-186*B*a)/a*(b*x+a)^(1/2))/b^5/x^5+7/256
*(9*A*b-10*B*a)/a^(11/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))

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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)/x^6/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.45123, size = 732, normalized size = 4.14 \begin{align*} \left [-\frac{105 \,{\left (10 \, B a b^{4} - 9 \, A b^{5}\right )} \sqrt{a} x^{5} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (384 \, A a^{5} - 105 \,{\left (10 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 70 \,{\left (10 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} - 56 \,{\left (10 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + 48 \,{\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt{b x + a}}{3840 \, a^{6} x^{5}}, \frac{105 \,{\left (10 \, B a b^{4} - 9 \, A b^{5}\right )} \sqrt{-a} x^{5} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) -{\left (384 \, A a^{5} - 105 \,{\left (10 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 70 \,{\left (10 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} - 56 \,{\left (10 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + 48 \,{\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt{b x + a}}{1920 \, a^{6} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^6/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/3840*(105*(10*B*a*b^4 - 9*A*b^5)*sqrt(a)*x^5*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(384*A*a^5 -
105*(10*B*a^2*b^3 - 9*A*a*b^4)*x^4 + 70*(10*B*a^3*b^2 - 9*A*a^2*b^3)*x^3 - 56*(10*B*a^4*b - 9*A*a^3*b^2)*x^2
+ 48*(10*B*a^5 - 9*A*a^4*b)*x)*sqrt(b*x + a))/(a^6*x^5), 1/1920*(105*(10*B*a*b^4 - 9*A*b^5)*sqrt(-a)*x^5*arcta
n(sqrt(b*x + a)*sqrt(-a)/a) - (384*A*a^5 - 105*(10*B*a^2*b^3 - 9*A*a*b^4)*x^4 + 70*(10*B*a^3*b^2 - 9*A*a^2*b^3
)*x^3 - 56*(10*B*a^4*b - 9*A*a^3*b^2)*x^2 + 48*(10*B*a^5 - 9*A*a^4*b)*x)*sqrt(b*x + a))/(a^6*x^5)]

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Sympy [B]  time = 177.493, size = 360, normalized size = 2.03 \begin{align*} - \frac{A}{5 \sqrt{b} x^{\frac{11}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{A \sqrt{b}}{40 a x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{3 A b^{\frac{3}{2}}}{80 a^{2} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{21 A b^{\frac{5}{2}}}{320 a^{3} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{21 A b^{\frac{7}{2}}}{128 a^{4} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{63 A b^{\frac{9}{2}}}{128 a^{5} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{63 A b^{5} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{128 a^{\frac{11}{2}}} - \frac{B}{4 \sqrt{b} x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{B \sqrt{b}}{24 a x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{7 B b^{\frac{3}{2}}}{96 a^{2} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{35 B b^{\frac{5}{2}}}{192 a^{3} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{35 B b^{\frac{7}{2}}}{64 a^{4} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{35 B b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{64 a^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**6/(b*x+a)**(1/2),x)

[Out]

-A/(5*sqrt(b)*x**(11/2)*sqrt(a/(b*x) + 1)) + A*sqrt(b)/(40*a*x**(9/2)*sqrt(a/(b*x) + 1)) - 3*A*b**(3/2)/(80*a*
*2*x**(7/2)*sqrt(a/(b*x) + 1)) + 21*A*b**(5/2)/(320*a**3*x**(5/2)*sqrt(a/(b*x) + 1)) - 21*A*b**(7/2)/(128*a**4
*x**(3/2)*sqrt(a/(b*x) + 1)) - 63*A*b**(9/2)/(128*a**5*sqrt(x)*sqrt(a/(b*x) + 1)) + 63*A*b**5*asinh(sqrt(a)/(s
qrt(b)*sqrt(x)))/(128*a**(11/2)) - B/(4*sqrt(b)*x**(9/2)*sqrt(a/(b*x) + 1)) + B*sqrt(b)/(24*a*x**(7/2)*sqrt(a/
(b*x) + 1)) - 7*B*b**(3/2)/(96*a**2*x**(5/2)*sqrt(a/(b*x) + 1)) + 35*B*b**(5/2)/(192*a**3*x**(3/2)*sqrt(a/(b*x
) + 1)) + 35*B*b**(7/2)/(64*a**4*sqrt(x)*sqrt(a/(b*x) + 1)) - 35*B*b**4*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(64*a
**(9/2))

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Giac [A]  time = 1.30303, size = 281, normalized size = 1.59 \begin{align*} \frac{\frac{105 \,{\left (10 \, B a b^{5} - 9 \, A b^{6}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{5}} + \frac{1050 \,{\left (b x + a\right )}^{\frac{9}{2}} B a b^{5} - 4900 \,{\left (b x + a\right )}^{\frac{7}{2}} B a^{2} b^{5} + 8960 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{3} b^{5} - 7900 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{4} b^{5} + 2790 \, \sqrt{b x + a} B a^{5} b^{5} - 945 \,{\left (b x + a\right )}^{\frac{9}{2}} A b^{6} + 4410 \,{\left (b x + a\right )}^{\frac{7}{2}} A a b^{6} - 8064 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{2} b^{6} + 7110 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{3} b^{6} - 2895 \, \sqrt{b x + a} A a^{4} b^{6}}{a^{5} b^{5} x^{5}}}{1920 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^6/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/1920*(105*(10*B*a*b^5 - 9*A*b^6)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^5) + (1050*(b*x + a)^(9/2)*B*a*b
^5 - 4900*(b*x + a)^(7/2)*B*a^2*b^5 + 8960*(b*x + a)^(5/2)*B*a^3*b^5 - 7900*(b*x + a)^(3/2)*B*a^4*b^5 + 2790*s
qrt(b*x + a)*B*a^5*b^5 - 945*(b*x + a)^(9/2)*A*b^6 + 4410*(b*x + a)^(7/2)*A*a*b^6 - 8064*(b*x + a)^(5/2)*A*a^2
*b^6 + 7110*(b*x + a)^(3/2)*A*a^3*b^6 - 2895*sqrt(b*x + a)*A*a^4*b^6)/(a^5*b^5*x^5))/b