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

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

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Rubi [A]  time = 0.08, antiderivative size = 177, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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 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 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*}

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Mathematica [C]  time = 0.03, size = 57, normalized size = 0.32 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(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]))/(a^6*x^5)

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IntegrateAlgebraic [A]  time = 0.27, size = 173, normalized size = 0.98 \begin {gather*} \frac {\sqrt {a+b x} \left (2790 a^5 B-2895 a^4 A b-7900 a^4 B (a+b x)+7110 a^3 A b (a+b x)+8960 a^3 B (a+b x)^2-8064 a^2 A b (a+b x)^2-4900 a^2 B (a+b x)^3+4410 a A b (a+b x)^3-945 A b (a+b x)^4+1050 a B (a+b x)^4\right )}{1920 a^5 b x^5}-\frac {7 \left (10 a b^4 B-9 A b^5\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*(-2895*a^4*A*b + 2790*a^5*B + 7110*a^3*A*b*(a + b*x) - 7900*a^4*B*(a + b*x) - 8064*a^2*A*b*(a +
b*x)^2 + 8960*a^3*B*(a + b*x)^2 + 4410*a*A*b*(a + b*x)^3 - 4900*a^2*B*(a + b*x)^3 - 945*A*b*(a + b*x)^4 + 105
0*a*B*(a + b*x)^4))/(1920*a^5*b*x^5) - (7*(-9*A*b^5 + 10*a*b^4*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(11/2
))

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fricas [A]  time = 1.13, size = 308, normalized size = 1.74 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.25, size = 208, normalized size = 1.18 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.02, size = 146, normalized size = 0.82 \begin {gather*} 2 \left (\frac {7 \left (9 A b -10 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {11}{2}}}+\frac {-\frac {\left (193 A b -186 B a \right ) \sqrt {b x +a}}{256 a}+\frac {79 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a^{2}}-\frac {7 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{30 a^{3}}+\frac {49 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{384 a^{4}}-\frac {7 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{5}}}{b^{5} x^{5}}\right ) b^{4} \end {gather*}

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))/x^5/b^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 [A]  time = 2.03, size = 233, normalized size = 1.32 \begin {gather*} \frac {1}{3840} \, b^{5} {\left (\frac {2 \, {\left (105 \, {\left (10 \, B a - 9 \, A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 490 \, {\left (10 \, B a^{2} - 9 \, A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 896 \, {\left (10 \, B a^{3} - 9 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 790 \, {\left (10 \, B a^{4} - 9 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (186 \, B a^{5} - 193 \, A a^{4} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{5} a^{5} b - 5 \, {\left (b x + a\right )}^{4} a^{6} b + 10 \, {\left (b x + a\right )}^{3} a^{7} b - 10 \, {\left (b x + a\right )}^{2} a^{8} b + 5 \, {\left (b x + a\right )} a^{9} b - a^{10} b} + \frac {105 \, {\left (10 \, B a - 9 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {11}{2}} b}\right )} \end {gather*}

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]

1/3840*b^5*(2*(105*(10*B*a - 9*A*b)*(b*x + a)^(9/2) - 490*(10*B*a^2 - 9*A*a*b)*(b*x + a)^(7/2) + 896*(10*B*a^3
- 9*A*a^2*b)*(b*x + a)^(5/2) - 790*(10*B*a^4 - 9*A*a^3*b)*(b*x + a)^(3/2) + 15*(186*B*a^5 - 193*A*a^4*b)*sqrt
(b*x + a))/((b*x + a)^5*a^5*b - 5*(b*x + a)^4*a^6*b + 10*(b*x + a)^3*a^7*b - 10*(b*x + a)^2*a^8*b + 5*(b*x + a
)*a^9*b - a^10*b) + 105*(10*B*a - 9*A*b)*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(11/2)*b)
)

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mupad [B]  time = 0.14, size = 221, normalized size = 1.25 \begin {gather*} \frac {\frac {7\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{5/2}}{15\,a^3}-\frac {79\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{3/2}}{192\,a^2}-\frac {49\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{7/2}}{192\,a^4}+\frac {7\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{9/2}}{128\,a^5}+\frac {\left (193\,A\,b^5-186\,B\,a\,b^4\right )\,\sqrt {a+b\,x}}{128\,a}}{5\,a\,{\left (a+b\,x\right )}^4-5\,a^4\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^5-10\,a^2\,{\left (a+b\,x\right )}^3+10\,a^3\,{\left (a+b\,x\right )}^2+a^5}+\frac {7\,b^4\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (9\,A\,b-10\,B\,a\right )}{128\,a^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((7*(9*A*b^5 - 10*B*a*b^4)*(a + b*x)^(5/2))/(15*a^3) - (79*(9*A*b^5 - 10*B*a*b^4)*(a + b*x)^(3/2))/(192*a^2) -
(49*(9*A*b^5 - 10*B*a*b^4)*(a + b*x)^(7/2))/(192*a^4) + (7*(9*A*b^5 - 10*B*a*b^4)*(a + b*x)^(9/2))/(128*a^5)
+ ((193*A*b^5 - 186*B*a*b^4)*(a + b*x)^(1/2))/(128*a))/(5*a*(a + b*x)^4 - 5*a^4*(a + b*x) - (a + b*x)^5 - 10*a
^2*(a + b*x)^3 + 10*a^3*(a + b*x)^2 + a^5) + (7*b^4*atanh((a + b*x)^(1/2)/a^(1/2))*(9*A*b - 10*B*a))/(128*a^(1
1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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