3.432 \(\int \frac{A+B x}{x^5 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=146 \[ \frac{5 b^2 \sqrt{a+b x} (7 A b-8 a B)}{64 a^4 x}-\frac{5 b^3 (7 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{9/2}}-\frac{5 b \sqrt{a+b x} (7 A b-8 a B)}{96 a^3 x^2}+\frac{\sqrt{a+b x} (7 A b-8 a B)}{24 a^2 x^3}-\frac{A \sqrt{a+b x}}{4 a x^4} \]

[Out]

-(A*Sqrt[a + b*x])/(4*a*x^4) + ((7*A*b - 8*a*B)*Sqrt[a + b*x])/(24*a^2*x^3) - (5*b*(7*A*b - 8*a*B)*Sqrt[a + b*
x])/(96*a^3*x^2) + (5*b^2*(7*A*b - 8*a*B)*Sqrt[a + b*x])/(64*a^4*x) - (5*b^3*(7*A*b - 8*a*B)*ArcTanh[Sqrt[a +
b*x]/Sqrt[a]])/(64*a^(9/2))

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Rubi [A]  time = 0.0629986, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 51, 63, 208} \[ \frac{5 b^2 \sqrt{a+b x} (7 A b-8 a B)}{64 a^4 x}-\frac{5 b^3 (7 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{9/2}}-\frac{5 b \sqrt{a+b x} (7 A b-8 a B)}{96 a^3 x^2}+\frac{\sqrt{a+b x} (7 A b-8 a B)}{24 a^2 x^3}-\frac{A \sqrt{a+b x}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^5*Sqrt[a + b*x]),x]

[Out]

-(A*Sqrt[a + b*x])/(4*a*x^4) + ((7*A*b - 8*a*B)*Sqrt[a + b*x])/(24*a^2*x^3) - (5*b*(7*A*b - 8*a*B)*Sqrt[a + b*
x])/(96*a^3*x^2) + (5*b^2*(7*A*b - 8*a*B)*Sqrt[a + b*x])/(64*a^4*x) - (5*b^3*(7*A*b - 8*a*B)*ArcTanh[Sqrt[a +
b*x]/Sqrt[a]])/(64*a^(9/2))

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^5 \sqrt{a+b x}} \, dx &=-\frac{A \sqrt{a+b x}}{4 a x^4}+\frac{\left (-\frac{7 A b}{2}+4 a B\right ) \int \frac{1}{x^4 \sqrt{a+b x}} \, dx}{4 a}\\ &=-\frac{A \sqrt{a+b x}}{4 a x^4}+\frac{(7 A b-8 a B) \sqrt{a+b x}}{24 a^2 x^3}+\frac{(5 b (7 A b-8 a B)) \int \frac{1}{x^3 \sqrt{a+b x}} \, dx}{48 a^2}\\ &=-\frac{A \sqrt{a+b x}}{4 a x^4}+\frac{(7 A b-8 a B) \sqrt{a+b x}}{24 a^2 x^3}-\frac{5 b (7 A b-8 a B) \sqrt{a+b x}}{96 a^3 x^2}-\frac{\left (5 b^2 (7 A b-8 a B)\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{64 a^3}\\ &=-\frac{A \sqrt{a+b x}}{4 a x^4}+\frac{(7 A b-8 a B) \sqrt{a+b x}}{24 a^2 x^3}-\frac{5 b (7 A b-8 a B) \sqrt{a+b x}}{96 a^3 x^2}+\frac{5 b^2 (7 A b-8 a B) \sqrt{a+b x}}{64 a^4 x}+\frac{\left (5 b^3 (7 A b-8 a B)\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{128 a^4}\\ &=-\frac{A \sqrt{a+b x}}{4 a x^4}+\frac{(7 A b-8 a B) \sqrt{a+b x}}{24 a^2 x^3}-\frac{5 b (7 A b-8 a B) \sqrt{a+b x}}{96 a^3 x^2}+\frac{5 b^2 (7 A b-8 a B) \sqrt{a+b x}}{64 a^4 x}+\frac{\left (5 b^2 (7 A b-8 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{64 a^4}\\ &=-\frac{A \sqrt{a+b x}}{4 a x^4}+\frac{(7 A b-8 a B) \sqrt{a+b x}}{24 a^2 x^3}-\frac{5 b (7 A b-8 a B) \sqrt{a+b x}}{96 a^3 x^2}+\frac{5 b^2 (7 A b-8 a B) \sqrt{a+b x}}{64 a^4 x}-\frac{5 b^3 (7 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0162654, size = 57, normalized size = 0.39 \[ -\frac{\sqrt{a+b x} \left (a^4 A+b^3 x^4 (7 A b-8 a B) \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{b x}{a}+1\right )\right )}{4 a^5 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + b*x]*(a^4*A + b^3*(7*A*b - 8*a*B)*x^4*Hypergeometric2F1[1/2, 4, 3/2, 1 + (b*x)/a]))/(4*a^5*x^4)

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Maple [A]  time = 0.011, size = 125, normalized size = 0.9 \begin{align*} 2\,{b}^{3} \left ({\frac{1}{{b}^{4}{x}^{4}} \left ({\frac{ \left ( 35\,Ab-40\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{128\,{a}^{4}}}-{\frac{ \left ( 385\,Ab-440\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{384\,{a}^{3}}}+{\frac{ \left ( 511\,Ab-584\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{384\,{a}^{2}}}-{\frac{ \left ( 93\,Ab-88\,Ba \right ) \sqrt{bx+a}}{128\,a}} \right ) }-{\frac{35\,Ab-40\,Ba}{128\,{a}^{9/2}}{\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^5/(b*x+a)^(1/2),x)

[Out]

2*b^3*((5/128*(7*A*b-8*B*a)/a^4*(b*x+a)^(7/2)-55/384/a^3*(7*A*b-8*B*a)*(b*x+a)^(5/2)+73/384/a^2*(7*A*b-8*B*a)*
(b*x+a)^(3/2)-1/128*(93*A*b-88*B*a)/a*(b*x+a)^(1/2))/b^4/x^4-5/128*(7*A*b-8*B*a)/a^(9/2)*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^5/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.51666, size = 606, normalized size = 4.15 \begin{align*} \left [-\frac{15 \,{\left (8 \, B a b^{3} - 7 \, A b^{4}\right )} \sqrt{a} x^{4} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (48 \, A a^{4} + 15 \,{\left (8 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} - 10 \,{\left (8 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt{b x + a}}{384 \, a^{5} x^{4}}, -\frac{15 \,{\left (8 \, B a b^{3} - 7 \, A b^{4}\right )} \sqrt{-a} x^{4} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (48 \, A a^{4} + 15 \,{\left (8 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} - 10 \,{\left (8 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt{b x + a}}{192 \, a^{5} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/384*(15*(8*B*a*b^3 - 7*A*b^4)*sqrt(a)*x^4*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(48*A*a^4 + 15*
(8*B*a^2*b^2 - 7*A*a*b^3)*x^3 - 10*(8*B*a^3*b - 7*A*a^2*b^2)*x^2 + 8*(8*B*a^4 - 7*A*a^3*b)*x)*sqrt(b*x + a))/(
a^5*x^4), -1/192*(15*(8*B*a*b^3 - 7*A*b^4)*sqrt(-a)*x^4*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (48*A*a^4 + 15*(8*B
*a^2*b^2 - 7*A*a*b^3)*x^3 - 10*(8*B*a^3*b - 7*A*a^2*b^2)*x^2 + 8*(8*B*a^4 - 7*A*a^3*b)*x)*sqrt(b*x + a))/(a^5*
x^4)]

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Sympy [B]  time = 117.454, size = 303, normalized size = 2.08 \begin{align*} - \frac{A}{4 \sqrt{b} x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{A \sqrt{b}}{24 a x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{7 A b^{\frac{3}{2}}}{96 a^{2} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{35 A b^{\frac{5}{2}}}{192 a^{3} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{35 A b^{\frac{7}{2}}}{64 a^{4} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{35 A b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{64 a^{\frac{9}{2}}} - \frac{B}{3 \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{B \sqrt{b}}{12 a x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 B b^{\frac{3}{2}}}{24 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 B b^{\frac{5}{2}}}{8 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{5 B b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 a^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**5/(b*x+a)**(1/2),x)

[Out]

-A/(4*sqrt(b)*x**(9/2)*sqrt(a/(b*x) + 1)) + A*sqrt(b)/(24*a*x**(7/2)*sqrt(a/(b*x) + 1)) - 7*A*b**(3/2)/(96*a**
2*x**(5/2)*sqrt(a/(b*x) + 1)) + 35*A*b**(5/2)/(192*a**3*x**(3/2)*sqrt(a/(b*x) + 1)) + 35*A*b**(7/2)/(64*a**4*s
qrt(x)*sqrt(a/(b*x) + 1)) - 35*A*b**4*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(64*a**(9/2)) - B/(3*sqrt(b)*x**(7/2)*s
qrt(a/(b*x) + 1)) + B*sqrt(b)/(12*a*x**(5/2)*sqrt(a/(b*x) + 1)) - 5*B*b**(3/2)/(24*a**2*x**(3/2)*sqrt(a/(b*x)
+ 1)) - 5*B*b**(5/2)/(8*a**3*sqrt(x)*sqrt(a/(b*x) + 1)) + 5*B*b**3*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(8*a**(7/2
))

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Giac [A]  time = 1.1838, size = 238, normalized size = 1.63 \begin{align*} -\frac{\frac{15 \,{\left (8 \, B a b^{4} - 7 \, A b^{5}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{120 \,{\left (b x + a\right )}^{\frac{7}{2}} B a b^{4} - 440 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{2} b^{4} + 584 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{3} b^{4} - 264 \, \sqrt{b x + a} B a^{4} b^{4} - 105 \,{\left (b x + a\right )}^{\frac{7}{2}} A b^{5} + 385 \,{\left (b x + a\right )}^{\frac{5}{2}} A a b^{5} - 511 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} b^{5} + 279 \, \sqrt{b x + a} A a^{3} b^{5}}{a^{4} b^{4} x^{4}}}{192 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/192*(15*(8*B*a*b^4 - 7*A*b^5)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + (120*(b*x + a)^(7/2)*B*a*b^4
- 440*(b*x + a)^(5/2)*B*a^2*b^4 + 584*(b*x + a)^(3/2)*B*a^3*b^4 - 264*sqrt(b*x + a)*B*a^4*b^4 - 105*(b*x + a)^
(7/2)*A*b^5 + 385*(b*x + a)^(5/2)*A*a*b^5 - 511*(b*x + a)^(3/2)*A*a^2*b^5 + 279*sqrt(b*x + a)*A*a^3*b^5)/(a^4*
b^4*x^4))/b