3.4.97 \(\int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx\) [397]

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

[Out]

-1/5*A*(b*x+a)^(3/2)/a/x^5-1/128*b^4*(7*A*b-10*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(9/2)+1/40*(7*A*b-10*B*a)
*(b*x+a)^(1/2)/a/x^4+1/240*b*(7*A*b-10*B*a)*(b*x+a)^(1/2)/a^2/x^3-1/192*b^2*(7*A*b-10*B*a)*(b*x+a)^(1/2)/a^3/x
^2+1/128*b^3*(7*A*b-10*B*a)*(b*x+a)^(1/2)/a^4/x

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Rubi [A]
time = 0.06, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 43, 44, 65, 214} \begin {gather*} -\frac {b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{9/2}}+\frac {b^3 \sqrt {a+b x} (7 A b-10 a B)}{128 a^4 x}-\frac {b^2 \sqrt {a+b x} (7 A b-10 a B)}{192 a^3 x^2}+\frac {b \sqrt {a+b x} (7 A b-10 a B)}{240 a^2 x^3}+\frac {\sqrt {a+b x} (7 A b-10 a B)}{40 a x^4}-\frac {A (a+b x)^{3/2}}{5 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(A + B*x))/x^6,x]

[Out]

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

Rule 43

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

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 79

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] || L
tQ[p, n]))))

Rule 214

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

Rubi steps

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

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Mathematica [A]
time = 0.30, size = 132, normalized size = 0.75 \begin {gather*} \frac {\frac {\sqrt {a} \sqrt {a+b x} \left (105 A b^4 x^4-16 a^3 b x (3 A+5 B x)-96 a^4 (4 A+5 B x)-10 a b^3 x^3 (7 A+15 B x)+4 a^2 b^2 x^2 (14 A+25 B x)\right )}{x^5}-15 b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1920 a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[a]*Sqrt[a + b*x]*(105*A*b^4*x^4 - 16*a^3*b*x*(3*A + 5*B*x) - 96*a^4*(4*A + 5*B*x) - 10*a*b^3*x^3*(7*A +
15*B*x) + 4*a^2*b^2*x^2*(14*A + 25*B*x)))/x^5 - 15*b^4*(7*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1920
*a^(9/2))

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Maple [A]
time = 0.07, size = 143, normalized size = 0.81

method	result	size
risch	\(-\frac {\sqrt {b x +a}\, \left (-105 A \,b^{4} x^{4}+150 B a \,b^{3} x^{4}+70 A a \,b^{3} x^{3}-100 B \,a^{2} b^{2} x^{3}-56 A \,a^{2} b^{2} x^{2}+80 B \,a^{3} b \,x^{2}+48 A \,a^{3} b x +480 B \,a^{4} x +384 A \,a^{4}\right )}{1920 x^{5} a^{4}}-\frac {b^{4} \left (7 A b -10 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {9}{2}}}\)	\(131\)
derivativedivides	\(2 b^{4} \left (-\frac {-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{4}}+\frac {7 \left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{384 a^{3}}-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{30 a^{2}}+\frac {\left (79 A b -58 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a}+\left (\frac {7 A b}{256}-\frac {5 B a}{128}\right ) \sqrt {b x +a}}{b^{5} x^{5}}-\frac {\left (7 A b -10 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {9}{2}}}\right )\)	\(143\)
default	\(2 b^{4} \left (-\frac {-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{4}}+\frac {7 \left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{384 a^{3}}-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{30 a^{2}}+\frac {\left (79 A b -58 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a}+\left (\frac {7 A b}{256}-\frac {5 B a}{128}\right ) \sqrt {b x +a}}{b^{5} x^{5}}-\frac {\left (7 A b -10 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {9}{2}}}\right )\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^4*(-(-1/256*(7*A*b-10*B*a)/a^4*(b*x+a)^(9/2)+7/384/a^3*(7*A*b-10*B*a)*(b*x+a)^(7/2)-1/30/a^2*(7*A*b-10*B*a
)*(b*x+a)^(5/2)+1/384*(79*A*b-58*B*a)/a*(b*x+a)^(3/2)+(7/256*A*b-5/128*B*a)*(b*x+a)^(1/2))/b^5/x^5-1/256*(7*A*
b-10*B*a)/a^(9/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))

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Maxima [A]
time = 0.49, size = 233, normalized size = 1.32 \begin {gather*} -\frac {1}{3840} \, b^{5} {\left (\frac {2 \, {\left (15 \, {\left (10 \, B a - 7 \, A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 70 \, {\left (10 \, B a^{2} - 7 \, A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 128 \, {\left (10 \, B a^{3} - 7 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (58 \, B a^{4} - 79 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - 15 \, {\left (10 \, B a^{5} - 7 \, A a^{4} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{5} a^{4} b - 5 \, {\left (b x + a\right )}^{4} a^{5} b + 10 \, {\left (b x + a\right )}^{3} a^{6} b - 10 \, {\left (b x + a\right )}^{2} a^{7} b + 5 \, {\left (b x + a\right )} a^{8} b - a^{9} b} + \frac {15 \, {\left (10 \, B a - 7 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}} b}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3840*b^5*(2*(15*(10*B*a - 7*A*b)*(b*x + a)^(9/2) - 70*(10*B*a^2 - 7*A*a*b)*(b*x + a)^(7/2) + 128*(10*B*a^3
- 7*A*a^2*b)*(b*x + a)^(5/2) - 10*(58*B*a^4 - 79*A*a^3*b)*(b*x + a)^(3/2) - 15*(10*B*a^5 - 7*A*a^4*b)*sqrt(b*x
+ a))/((b*x + a)^5*a^4*b - 5*(b*x + a)^4*a^5*b + 10*(b*x + a)^3*a^6*b - 10*(b*x + a)^2*a^7*b + 5*(b*x + a)*a^
8*b - a^9*b) + 15*(10*B*a - 7*A*b)*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(9/2)*b))

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Fricas [A]
time = 1.12, size = 305, normalized size = 1.72 \begin {gather*} \left [-\frac {15 \, {\left (10 \, B a b^{4} - 7 \, 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} + 15 \, {\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{4} - 10 \, {\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{3} + 8 \, {\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} + A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3840 \, a^{5} x^{5}}, -\frac {15 \, {\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (384 \, A a^{5} + 15 \, {\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{4} - 10 \, {\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{3} + 8 \, {\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} + A a^{4} b\right )} x\right )} \sqrt {b x + a}}{1920 \, a^{5} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/3840*(15*(10*B*a*b^4 - 7*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 +
15*(10*B*a^2*b^3 - 7*A*a*b^4)*x^4 - 10*(10*B*a^3*b^2 - 7*A*a^2*b^3)*x^3 + 8*(10*B*a^4*b - 7*A*a^3*b^2)*x^2 + 4
8*(10*B*a^5 + A*a^4*b)*x)*sqrt(b*x + a))/(a^5*x^5), -1/1920*(15*(10*B*a*b^4 - 7*A*b^5)*sqrt(-a)*x^5*arctan(sqr
t(b*x + a)*sqrt(-a)/a) + (384*A*a^5 + 15*(10*B*a^2*b^3 - 7*A*a*b^4)*x^4 - 10*(10*B*a^3*b^2 - 7*A*a^2*b^3)*x^3
+ 8*(10*B*a^4*b - 7*A*a^3*b^2)*x^2 + 48*(10*B*a^5 + A*a^4*b)*x)*sqrt(b*x + a))/(a^5*x^5)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1416 vs. \(2 (165) = 330\).
time = 22.59, size = 1416, normalized size = 8.00 \begin {gather*} - \frac {1930 A a^{5} b^{5} \sqrt {a + b x}}{5120 a^{10} + 6400 a^{9} b x - 12800 a^{8} \left (a + b x\right )^{2} + 12800 a^{7} \left (a + b x\right )^{3} - 6400 a^{6} \left (a + b x\right )^{4} + 1280 a^{5} \left (a + b x\right )^{5}} + \frac {4740 A a^{4} b^{5} \left (a + b x\right )^{\frac {3}{2}}}{5120 a^{10} + 6400 a^{9} b x - 12800 a^{8} \left (a + b x\right )^{2} + 12800 a^{7} \left (a + b x\right )^{3} - 6400 a^{6} \left (a + b x\right )^{4} + 1280 a^{5} \left (a + b x\right )^{5}} - \frac {5376 A a^{3} b^{5} \left (a + b x\right )^{\frac {5}{2}}}{5120 a^{10} + 6400 a^{9} b x - 12800 a^{8} \left (a + b x\right )^{2} + 12800 a^{7} \left (a + b x\right )^{3} - 6400 a^{6} \left (a + b x\right )^{4} + 1280 a^{5} \left (a + b x\right )^{5}} - \frac {558 A a^{3} b^{5} \sqrt {a + b x}}{- 1152 a^{8} - 1536 a^{7} b x + 2304 a^{6} \left (a + b x\right )^{2} - 1536 a^{5} \left (a + b x\right )^{3} + 384 a^{4} \left (a + b x\right )^{4}} + \frac {2940 A a^{2} b^{5} \left (a + b x\right )^{\frac {7}{2}}}{5120 a^{10} + 6400 a^{9} b x - 12800 a^{8} \left (a + b x\right )^{2} + 12800 a^{7} \left (a + b x\right )^{3} - 6400 a^{6} \left (a + b x\right )^{4} + 1280 a^{5} \left (a + b x\right )^{5}} + \frac {1022 A a^{2} b^{5} \left (a + b x\right )^{\frac {3}{2}}}{- 1152 a^{8} - 1536 a^{7} b x + 2304 a^{6} \left (a + b x\right )^{2} - 1536 a^{5} \left (a + b x\right )^{3} + 384 a^{4} \left (a + b x\right )^{4}} - \frac {630 A a b^{5} \left (a + b x\right )^{\frac {9}{2}}}{5120 a^{10} + 6400 a^{9} b x - 12800 a^{8} \left (a + b x\right )^{2} + 12800 a^{7} \left (a + b x\right )^{3} - 6400 a^{6} \left (a + b x\right )^{4} + 1280 a^{5} \left (a + b x\right )^{5}} - \frac {770 A a b^{5} \left (a + b x\right )^{\frac {5}{2}}}{- 1152 a^{8} - 1536 a^{7} b x + 2304 a^{6} \left (a + b x\right )^{2} - 1536 a^{5} \left (a + b x\right )^{3} + 384 a^{4} \left (a + b x\right )^{4}} - \frac {63 A a b^{5} \sqrt {\frac {1}{a^{11}}} \log {\left (- a^{6} \sqrt {\frac {1}{a^{11}}} + \sqrt {a + b x} \right )}}{256} + \frac {63 A a b^{5} \sqrt {\frac {1}{a^{11}}} \log {\left (a^{6} \sqrt {\frac {1}{a^{11}}} + \sqrt {a + b x} \right )}}{256} + \frac {210 A b^{5} \left (a + b x\right )^{\frac {7}{2}}}{- 1152 a^{8} - 1536 a^{7} b x + 2304 a^{6} \left (a + b x\right )^{2} - 1536 a^{5} \left (a + b x\right )^{3} + 384 a^{4} \left (a + b x\right )^{4}} + \frac {35 A b^{5} \sqrt {\frac {1}{a^{9}}} \log {\left (- a^{5} \sqrt {\frac {1}{a^{9}}} + \sqrt {a + b x} \right )}}{128} - \frac {35 A b^{5} \sqrt {\frac {1}{a^{9}}} \log {\left (a^{5} \sqrt {\frac {1}{a^{9}}} + \sqrt {a + b x} \right )}}{128} - \frac {558 B a^{4} b^{4} \sqrt {a + b x}}{- 1152 a^{8} - 1536 a^{7} b x + 2304 a^{6} \left (a + b x\right )^{2} - 1536 a^{5} \left (a + b x\right )^{3} + 384 a^{4} \left (a + b x\right )^{4}} + \frac {1022 B a^{3} b^{4} \left (a + b x\right )^{\frac {3}{2}}}{- 1152 a^{8} - 1536 a^{7} b x + 2304 a^{6} \left (a + b x\right )^{2} - 1536 a^{5} \left (a + b x\right )^{3} + 384 a^{4} \left (a + b x\right )^{4}} - \frac {770 B a^{2} b^{4} \left (a + b x\right )^{\frac {5}{2}}}{- 1152 a^{8} - 1536 a^{7} b x + 2304 a^{6} \left (a + b x\right )^{2} - 1536 a^{5} \left (a + b x\right )^{3} + 384 a^{4} \left (a + b x\right )^{4}} - \frac {66 B a^{2} b^{4} \sqrt {a + b x}}{96 a^{6} + 144 a^{5} b x - 144 a^{4} \left (a + b x\right )^{2} + 48 a^{3} \left (a + b x\right )^{3}} + \frac {210 B a b^{4} \left (a + b x\right )^{\frac {7}{2}}}{- 1152 a^{8} - 1536 a^{7} b x + 2304 a^{6} \left (a + b x\right )^{2} - 1536 a^{5} \left (a + b x\right )^{3} + 384 a^{4} \left (a + b x\right )^{4}} + \frac {80 B a b^{4} \left (a + b x\right )^{\frac {3}{2}}}{96 a^{6} + 144 a^{5} b x - 144 a^{4} \left (a + b x\right )^{2} + 48 a^{3} \left (a + b x\right )^{3}} + \frac {35 B a b^{4} \sqrt {\frac {1}{a^{9}}} \log {\left (- a^{5} \sqrt {\frac {1}{a^{9}}} + \sqrt {a + b x} \right )}}{128} - \frac {35 B a b^{4} \sqrt {\frac {1}{a^{9}}} \log {\left (a^{5} \sqrt {\frac {1}{a^{9}}} + \sqrt {a + b x} \right )}}{128} - \frac {30 B b^{4} \left (a + b x\right )^{\frac {5}{2}}}{96 a^{6} + 144 a^{5} b x - 144 a^{4} \left (a + b x\right )^{2} + 48 a^{3} \left (a + b x\right )^{3}} - \frac {5 B b^{4} \sqrt {\frac {1}{a^{7}}} \log {\left (- a^{4} \sqrt {\frac {1}{a^{7}}} + \sqrt {a + b x} \right )}}{16} + \frac {5 B b^{4} \sqrt {\frac {1}{a^{7}}} \log {\left (a^{4} \sqrt {\frac {1}{a^{7}}} + \sqrt {a + b x} \right )}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1930*A*a**5*b**5*sqrt(a + b*x)/(5120*a**10 + 6400*a**9*b*x - 12800*a**8*(a + b*x)**2 + 12800*a**7*(a + b*x)**
3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)**5) + 4740*A*a**4*b**5*(a + b*x)**(3/2)/(5120*a**10 + 6400*a*
*9*b*x - 12800*a**8*(a + b*x)**2 + 12800*a**7*(a + b*x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)**5)
- 5376*A*a**3*b**5*(a + b*x)**(5/2)/(5120*a**10 + 6400*a**9*b*x - 12800*a**8*(a + b*x)**2 + 12800*a**7*(a + b*
x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)**5) - 558*A*a**3*b**5*sqrt(a + b*x)/(-1152*a**8 - 1536*a*
*7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) + 2940*A*a**2*b**5*(a + b*x)
**(7/2)/(5120*a**10 + 6400*a**9*b*x - 12800*a**8*(a + b*x)**2 + 12800*a**7*(a + b*x)**3 - 6400*a**6*(a + b*x)*
*4 + 1280*a**5*(a + b*x)**5) + 1022*A*a**2*b**5*(a + b*x)**(3/2)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a +
b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) - 630*A*a*b**5*(a + b*x)**(9/2)/(5120*a**10 + 6400*a
**9*b*x - 12800*a**8*(a + b*x)**2 + 12800*a**7*(a + b*x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)**5)
- 770*A*a*b**5*(a + b*x)**(5/2)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3
+ 384*a**4*(a + b*x)**4) - 63*A*a*b**5*sqrt(a**(-11))*log(-a**6*sqrt(a**(-11)) + sqrt(a + b*x))/256 + 63*A*a*
b**5*sqrt(a**(-11))*log(a**6*sqrt(a**(-11)) + sqrt(a + b*x))/256 + 210*A*b**5*(a + b*x)**(7/2)/(-1152*a**8 - 1
536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) + 35*A*b**5*sqrt(a**(-
9))*log(-a**5*sqrt(a**(-9)) + sqrt(a + b*x))/128 - 35*A*b**5*sqrt(a**(-9))*log(a**5*sqrt(a**(-9)) + sqrt(a + b
*x))/128 - 558*B*a**4*b**4*sqrt(a + b*x)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a +
b*x)**3 + 384*a**4*(a + b*x)**4) + 1022*B*a**3*b**4*(a + b*x)**(3/2)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*
(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) - 770*B*a**2*b**4*(a + b*x)**(5/2)/(-1152*a**8
- 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) - 66*B*a**2*b**4*sq
rt(a + b*x)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) + 210*B*a*b**4*(a + b*x)**
(7/2)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) +
80*B*a*b**4*(a + b*x)**(3/2)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) + 35*B*a
*b**4*sqrt(a**(-9))*log(-a**5*sqrt(a**(-9)) + sqrt(a + b*x))/128 - 35*B*a*b**4*sqrt(a**(-9))*log(a**5*sqrt(a**
(-9)) + sqrt(a + b*x))/128 - 30*B*b**4*(a + b*x)**(5/2)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a
**3*(a + b*x)**3) - 5*B*b**4*sqrt(a**(-7))*log(-a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 5*B*b**4*sqrt(a**(-7)
)*log(a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16

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Giac [A]
time = 1.79, size = 208, normalized size = 1.18 \begin {gather*} -\frac {\frac {15 \, {\left (10 \, B a b^{5} - 7 \, A b^{6}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {150 \, {\left (b x + a\right )}^{\frac {9}{2}} B a b^{5} - 700 \, {\left (b x + a\right )}^{\frac {7}{2}} B a^{2} b^{5} + 1280 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{3} b^{5} - 580 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{4} b^{5} - 150 \, \sqrt {b x + a} B a^{5} b^{5} - 105 \, {\left (b x + a\right )}^{\frac {9}{2}} A b^{6} + 490 \, {\left (b x + a\right )}^{\frac {7}{2}} A a b^{6} - 896 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{2} b^{6} + 790 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{3} b^{6} + 105 \, \sqrt {b x + a} A a^{4} b^{6}}{a^{4} b^{5} x^{5}}}{1920 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1920*(15*(10*B*a*b^5 - 7*A*b^6)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + (150*(b*x + a)^(9/2)*B*a*b^
5 - 700*(b*x + a)^(7/2)*B*a^2*b^5 + 1280*(b*x + a)^(5/2)*B*a^3*b^5 - 580*(b*x + a)^(3/2)*B*a^4*b^5 - 150*sqrt(
b*x + a)*B*a^5*b^5 - 105*(b*x + a)^(9/2)*A*b^6 + 490*(b*x + a)^(7/2)*A*a*b^6 - 896*(b*x + a)^(5/2)*A*a^2*b^6 +
790*(b*x + a)^(3/2)*A*a^3*b^6 + 105*sqrt(b*x + a)*A*a^4*b^6)/(a^4*b^5*x^5))/b

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Mupad [B]
time = 0.45, size = 217, normalized size = 1.23 \begin {gather*} \frac {\left (\frac {7\,A\,b^5}{128}-\frac {5\,B\,a\,b^4}{64}\right )\,\sqrt {a+b\,x}-\frac {\left (7\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{5/2}}{15\,a^2}+\frac {7\,\left (7\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{7/2}}{192\,a^3}-\frac {\left (7\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{9/2}}{128\,a^4}+\frac {\left (79\,A\,b^5-58\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{3/2}}{192\,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 {b^4\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (7\,A\,b-10\,B\,a\right )}{128\,a^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((7*A*b^5)/128 - (5*B*a*b^4)/64)*(a + b*x)^(1/2) - ((7*A*b^5 - 10*B*a*b^4)*(a + b*x)^(5/2))/(15*a^2) + (7*(7*
A*b^5 - 10*B*a*b^4)*(a + b*x)^(7/2))/(192*a^3) - ((7*A*b^5 - 10*B*a*b^4)*(a + b*x)^(9/2))/(128*a^4) + ((79*A*b
^5 - 58*B*a*b^4)*(a + b*x)^(3/2))/(192*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) - (b^4*atanh((a + b*x)^(1/2)/a^(1/2))*(7*A*b - 10*B*a))/(128*a^(9/2))

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