\(\int \frac {(a+b x)^{5/2} (A+B x)}{x^8} \, dx\) [422]

Optimal result

Integrand size = 18, antiderivative size = 232 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^8} \, dx=\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b^3 (A b-2 a B) \sqrt {a+b x}}{384 a^2 x^3}-\frac {5 b^4 (A b-2 a B) \sqrt {a+b x}}{1536 a^3 x^2}+\frac {5 b^5 (A b-2 a B) \sqrt {a+b x}}{1024 a^4 x}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}-\frac {5 b^6 (A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{9/2}} \]

[Out]

1/24*b*(A*b-2*B*a)*(b*x+a)^(3/2)/a/x^5+1/12*(A*b-2*B*a)*(b*x+a)^(5/2)/a/x^6-1/7*A*(b*x+a)^(7/2)/a/x^7-5/1024*b
^6*(A*b-2*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(9/2)+1/64*b^2*(A*b-2*B*a)*(b*x+a)^(1/2)/a/x^4+1/384*b^3*(A*b-
2*B*a)*(b*x+a)^(1/2)/a^2/x^3-5/1536*b^4*(A*b-2*B*a)*(b*x+a)^(1/2)/a^3/x^2+5/1024*b^5*(A*b-2*B*a)*(b*x+a)^(1/2)
/a^4/x

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 43, 44, 65, 214} \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^8} \, dx=-\frac {5 b^6 (A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{9/2}}+\frac {5 b^5 \sqrt {a+b x} (A b-2 a B)}{1024 a^4 x}-\frac {5 b^4 \sqrt {a+b x} (A b-2 a B)}{1536 a^3 x^2}+\frac {b^3 \sqrt {a+b x} (A b-2 a B)}{384 a^2 x^3}+\frac {b^2 \sqrt {a+b x} (A b-2 a B)}{64 a x^4}+\frac {(a+b x)^{5/2} (A b-2 a B)}{12 a x^6}+\frac {b (a+b x)^{3/2} (A b-2 a B)}{24 a x^5}-\frac {A (a+b x)^{7/2}}{7 a x^7} \]

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/x^8,x]

[Out]

(b^2*(A*b - 2*a*B)*Sqrt[a + b*x])/(64*a*x^4) + (b^3*(A*b - 2*a*B)*Sqrt[a + b*x])/(384*a^2*x^3) - (5*b^4*(A*b -
2*a*B)*Sqrt[a + b*x])/(1536*a^3*x^2) + (5*b^5*(A*b - 2*a*B)*Sqrt[a + b*x])/(1024*a^4*x) + (b*(A*b - 2*a*B)*(a
+ b*x)^(3/2))/(24*a*x^5) + ((A*b - 2*a*B)*(a + b*x)^(5/2))/(12*a*x^6) - (A*(a + b*x)^(7/2))/(7*a*x^7) - (5*b^
6*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*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*} \text {integral}& = -\frac {A (a+b x)^{7/2}}{7 a x^7}+\frac {\left (-\frac {7 A b}{2}+7 a B\right ) \int \frac {(a+b x)^{5/2}}{x^7} \, dx}{7 a} \\ & = \frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}-\frac {(5 b (A b-2 a B)) \int \frac {(a+b x)^{3/2}}{x^6} \, dx}{24 a} \\ & = \frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}-\frac {\left (b^2 (A b-2 a B)\right ) \int \frac {\sqrt {a+b x}}{x^5} \, dx}{16 a} \\ & = \frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}-\frac {\left (b^3 (A b-2 a B)\right ) \int \frac {1}{x^4 \sqrt {a+b x}} \, dx}{128 a} \\ & = \frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b^3 (A b-2 a B) \sqrt {a+b x}}{384 a^2 x^3}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}+\frac {\left (5 b^4 (A b-2 a B)\right ) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{768 a^2} \\ & = \frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b^3 (A b-2 a B) \sqrt {a+b x}}{384 a^2 x^3}-\frac {5 b^4 (A b-2 a B) \sqrt {a+b x}}{1536 a^3 x^2}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}-\frac {\left (5 b^5 (A b-2 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{1024 a^3} \\ & = \frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b^3 (A b-2 a B) \sqrt {a+b x}}{384 a^2 x^3}-\frac {5 b^4 (A b-2 a B) \sqrt {a+b x}}{1536 a^3 x^2}+\frac {5 b^5 (A b-2 a B) \sqrt {a+b x}}{1024 a^4 x}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}+\frac {\left (5 b^6 (A b-2 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{2048 a^4} \\ & = \frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b^3 (A b-2 a B) \sqrt {a+b x}}{384 a^2 x^3}-\frac {5 b^4 (A b-2 a B) \sqrt {a+b x}}{1536 a^3 x^2}+\frac {5 b^5 (A b-2 a B) \sqrt {a+b x}}{1024 a^4 x}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}+\frac {\left (5 b^5 (A b-2 a B)\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{1024 a^4} \\ & = \frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b^3 (A b-2 a B) \sqrt {a+b x}}{384 a^2 x^3}-\frac {5 b^4 (A b-2 a B) \sqrt {a+b x}}{1536 a^3 x^2}+\frac {5 b^5 (A b-2 a B) \sqrt {a+b x}}{1024 a^4 x}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}-\frac {5 b^6 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.72 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^8} \, dx=-\frac {\sqrt {a+b x} \left (-105 A b^6 x^6+70 a b^5 x^5 (A+3 B x)-28 a^2 b^4 x^4 (2 A+5 B x)+16 a^3 b^3 x^3 (3 A+7 B x)+512 a^6 (6 A+7 B x)+256 a^5 b x (29 A+35 B x)+32 a^4 b^2 x^2 (148 A+189 B x)\right )}{21504 a^4 x^7}+\frac {5 b^6 (-A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{9/2}} \]

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/x^8,x]

[Out]

-1/21504*(Sqrt[a + b*x]*(-105*A*b^6*x^6 + 70*a*b^5*x^5*(A + 3*B*x) - 28*a^2*b^4*x^4*(2*A + 5*B*x) + 16*a^3*b^3
*x^3*(3*A + 7*B*x) + 512*a^6*(6*A + 7*B*x) + 256*a^5*b*x*(29*A + 35*B*x) + 32*a^4*b^2*x^2*(148*A + 189*B*x)))/
(a^4*x^7) + (5*b^6*(-(A*b) + 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(9/2))

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.65

method	result	size
pseudoelliptic	\(\frac {-\frac {15 b^{6} x^{7} \left (A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8}+\sqrt {b x +a}\, \left (-\frac {928 x b \left (\frac {35 B x}{29}+A \right ) a^{\frac {11}{2}}}{7}+\left (-64 B x -\frac {384 A}{7}\right ) a^{\frac {13}{2}}+x^{2} b^{2} \left (-\frac {5 b^{3} x^{3} \left (3 B x +A \right ) a^{\frac {3}{2}}}{4}+b^{2} x^{2} \left (\frac {5 B x}{2}+A \right ) a^{\frac {5}{2}}-\frac {6 x \left (\frac {7 B x}{3}+A \right ) b \,a^{\frac {7}{2}}}{7}+\left (-108 B x -\frac {592 A}{7}\right ) a^{\frac {9}{2}}+\frac {15 A \sqrt {a}\, b^{4} x^{4}}{8}\right )\right )}{384 a^{\frac {9}{2}} x^{7}}\)	\(150\)
derivativedivides	\(2 b^{6} \left (-\frac {-\frac {5 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {13}{2}}}{2048 a^{4}}+\frac {25 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1536 a^{3}}-\frac {283 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{6144 a^{2}}+\frac {A b \left (b x +a \right )^{\frac {7}{2}}}{14 a}+\left (\frac {283 A b}{6144}-\frac {283 B a}{3072}\right ) \left (b x +a \right )^{\frac {5}{2}}-\frac {25 a \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{1536}+\frac {5 a^{2} \left (A b -2 B a \right ) \sqrt {b x +a}}{2048}}{b^{7} x^{7}}-\frac {5 \left (A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2048 a^{\frac {9}{2}}}\right )\)	\(170\)
default	\(2 b^{6} \left (-\frac {-\frac {5 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {13}{2}}}{2048 a^{4}}+\frac {25 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1536 a^{3}}-\frac {283 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{6144 a^{2}}+\frac {A b \left (b x +a \right )^{\frac {7}{2}}}{14 a}+\left (\frac {283 A b}{6144}-\frac {283 B a}{3072}\right ) \left (b x +a \right )^{\frac {5}{2}}-\frac {25 a \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{1536}+\frac {5 a^{2} \left (A b -2 B a \right ) \sqrt {b x +a}}{2048}}{b^{7} x^{7}}-\frac {5 \left (A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2048 a^{\frac {9}{2}}}\right )\)	\(170\)
risch	\(-\frac {\sqrt {b x +a}\, \left (-105 A \,b^{6} x^{6}+210 B a \,b^{5} x^{6}+70 A a \,b^{5} x^{5}-140 B \,a^{2} b^{4} x^{5}-56 A \,a^{2} b^{4} x^{4}+112 B \,a^{3} b^{3} x^{4}+48 A \,a^{3} b^{3} x^{3}+6048 B \,a^{4} b^{2} x^{3}+4736 A \,a^{4} b^{2} x^{2}+8960 B \,a^{5} b \,x^{2}+7424 A \,a^{5} b x +3584 B \,a^{6} x +3072 A \,a^{6}\right )}{21504 x^{7} a^{4}}-\frac {5 b^{6} \left (A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {9}{2}}}\)	\(178\)

[In]

int((b*x+a)^(5/2)*(B*x+A)/x^8,x,method=_RETURNVERBOSE)

[Out]

1/384*(-15/8*b^6*x^7*(A*b-2*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))+(b*x+a)^(1/2)*(-928/7*x*b*(35/29*B*x+A)*a^(11/
2)+(-64*B*x-384/7*A)*a^(13/2)+x^2*b^2*(-5/4*b^3*x^3*(3*B*x+A)*a^(3/2)+b^2*x^2*(5/2*B*x+A)*a^(5/2)-6/7*x*(7/3*B
*x+A)*b*a^(7/2)+(-108*B*x-592/7*A)*a^(9/2)+15/8*A*a^(1/2)*b^4*x^4)))/a^(9/2)/x^7

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.73 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^8} \, dx=\left [-\frac {105 \, {\left (2 \, B a b^{6} - A b^{7}\right )} \sqrt {a} x^{7} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3072 \, A a^{7} + 105 \, {\left (2 \, B a^{2} b^{5} - A a b^{6}\right )} x^{6} - 70 \, {\left (2 \, B a^{3} b^{4} - A a^{2} b^{5}\right )} x^{5} + 56 \, {\left (2 \, B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{4} + 48 \, {\left (126 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{3} + 128 \, {\left (70 \, B a^{6} b + 37 \, A a^{5} b^{2}\right )} x^{2} + 256 \, {\left (14 \, B a^{7} + 29 \, A a^{6} b\right )} x\right )} \sqrt {b x + a}}{43008 \, a^{5} x^{7}}, -\frac {105 \, {\left (2 \, B a b^{6} - A b^{7}\right )} \sqrt {-a} x^{7} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (3072 \, A a^{7} + 105 \, {\left (2 \, B a^{2} b^{5} - A a b^{6}\right )} x^{6} - 70 \, {\left (2 \, B a^{3} b^{4} - A a^{2} b^{5}\right )} x^{5} + 56 \, {\left (2 \, B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{4} + 48 \, {\left (126 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{3} + 128 \, {\left (70 \, B a^{6} b + 37 \, A a^{5} b^{2}\right )} x^{2} + 256 \, {\left (14 \, B a^{7} + 29 \, A a^{6} b\right )} x\right )} \sqrt {b x + a}}{21504 \, a^{5} x^{7}}\right ] \]

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^8,x, algorithm="fricas")

[Out]

[-1/43008*(105*(2*B*a*b^6 - A*b^7)*sqrt(a)*x^7*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(3072*A*a^7 +
105*(2*B*a^2*b^5 - A*a*b^6)*x^6 - 70*(2*B*a^3*b^4 - A*a^2*b^5)*x^5 + 56*(2*B*a^4*b^3 - A*a^3*b^4)*x^4 + 48*(12
6*B*a^5*b^2 + A*a^4*b^3)*x^3 + 128*(70*B*a^6*b + 37*A*a^5*b^2)*x^2 + 256*(14*B*a^7 + 29*A*a^6*b)*x)*sqrt(b*x +
a))/(a^5*x^7), -1/21504*(105*(2*B*a*b^6 - A*b^7)*sqrt(-a)*x^7*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (3072*A*a^7
+ 105*(2*B*a^2*b^5 - A*a*b^6)*x^6 - 70*(2*B*a^3*b^4 - A*a^2*b^5)*x^5 + 56*(2*B*a^4*b^3 - A*a^3*b^4)*x^4 + 48*(
126*B*a^5*b^2 + A*a^4*b^3)*x^3 + 128*(70*B*a^6*b + 37*A*a^5*b^2)*x^2 + 256*(14*B*a^7 + 29*A*a^6*b)*x)*sqrt(b*x
+ a))/(a^5*x^7)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^8} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**(5/2)*(B*x+A)/x**8,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^8} \, dx=-\frac {1}{43008} \, b^{7} {\left (\frac {2 \, {\left (3072 \, {\left (b x + a\right )}^{\frac {7}{2}} A a^{3} b + 105 \, {\left (2 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {13}{2}} - 700 \, {\left (2 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {11}{2}} + 1981 \, {\left (2 \, B a^{3} - A a^{2} b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 1981 \, {\left (2 \, B a^{5} - A a^{4} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 700 \, {\left (2 \, B a^{6} - A a^{5} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - 105 \, {\left (2 \, B a^{7} - A a^{6} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{7} a^{4} b - 7 \, {\left (b x + a\right )}^{6} a^{5} b + 21 \, {\left (b x + a\right )}^{5} a^{6} b - 35 \, {\left (b x + a\right )}^{4} a^{7} b + 35 \, {\left (b x + a\right )}^{3} a^{8} b - 21 \, {\left (b x + a\right )}^{2} a^{9} b + 7 \, {\left (b x + a\right )} a^{10} b - a^{11} b} + \frac {105 \, {\left (2 \, B a - A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}} b}\right )} \]

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^8,x, algorithm="maxima")

[Out]

-1/43008*b^7*(2*(3072*(b*x + a)^(7/2)*A*a^3*b + 105*(2*B*a - A*b)*(b*x + a)^(13/2) - 700*(2*B*a^2 - A*a*b)*(b*
x + a)^(11/2) + 1981*(2*B*a^3 - A*a^2*b)*(b*x + a)^(9/2) - 1981*(2*B*a^5 - A*a^4*b)*(b*x + a)^(5/2) + 700*(2*B
*a^6 - A*a^5*b)*(b*x + a)^(3/2) - 105*(2*B*a^7 - A*a^6*b)*sqrt(b*x + a))/((b*x + a)^7*a^4*b - 7*(b*x + a)^6*a^
5*b + 21*(b*x + a)^5*a^6*b - 35*(b*x + a)^4*a^7*b + 35*(b*x + a)^3*a^8*b - 21*(b*x + a)^2*a^9*b + 7*(b*x + a)*
a^10*b - a^11*b) + 105*(2*B*a - A*b)*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(9/2)*b))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^8} \, dx=-\frac {\frac {105 \, {\left (2 \, B a b^{7} - A b^{8}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {210 \, {\left (b x + a\right )}^{\frac {13}{2}} B a b^{7} - 1400 \, {\left (b x + a\right )}^{\frac {11}{2}} B a^{2} b^{7} + 3962 \, {\left (b x + a\right )}^{\frac {9}{2}} B a^{3} b^{7} - 3962 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{5} b^{7} + 1400 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{6} b^{7} - 210 \, \sqrt {b x + a} B a^{7} b^{7} - 105 \, {\left (b x + a\right )}^{\frac {13}{2}} A b^{8} + 700 \, {\left (b x + a\right )}^{\frac {11}{2}} A a b^{8} - 1981 \, {\left (b x + a\right )}^{\frac {9}{2}} A a^{2} b^{8} + 3072 \, {\left (b x + a\right )}^{\frac {7}{2}} A a^{3} b^{8} + 1981 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{4} b^{8} - 700 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{5} b^{8} + 105 \, \sqrt {b x + a} A a^{6} b^{8}}{a^{4} b^{7} x^{7}}}{21504 \, b} \]

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^8,x, algorithm="giac")

[Out]

-1/21504*(105*(2*B*a*b^7 - A*b^8)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + (210*(b*x + a)^(13/2)*B*a*b^
7 - 1400*(b*x + a)^(11/2)*B*a^2*b^7 + 3962*(b*x + a)^(9/2)*B*a^3*b^7 - 3962*(b*x + a)^(5/2)*B*a^5*b^7 + 1400*(
b*x + a)^(3/2)*B*a^6*b^7 - 210*sqrt(b*x + a)*B*a^7*b^7 - 105*(b*x + a)^(13/2)*A*b^8 + 700*(b*x + a)^(11/2)*A*a
*b^8 - 1981*(b*x + a)^(9/2)*A*a^2*b^8 + 3072*(b*x + a)^(7/2)*A*a^3*b^8 + 1981*(b*x + a)^(5/2)*A*a^4*b^8 - 700*
(b*x + a)^(3/2)*A*a^5*b^8 + 105*sqrt(b*x + a)*A*a^6*b^8)/(a^4*b^7*x^7))/b

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^8} \, dx=\frac {\left (\frac {283\,A\,b^7}{3072}-\frac {283\,B\,a\,b^6}{1536}\right )\,{\left (a+b\,x\right )}^{5/2}+\left (\frac {5\,A\,a^2\,b^7}{1024}-\frac {5\,B\,a^3\,b^6}{512}\right )\,\sqrt {a+b\,x}+\left (\frac {25\,B\,a^2\,b^6}{384}-\frac {25\,A\,a\,b^7}{768}\right )\,{\left (a+b\,x\right )}^{3/2}-\frac {283\,\left (A\,b^7-2\,B\,a\,b^6\right )\,{\left (a+b\,x\right )}^{9/2}}{3072\,a^2}+\frac {25\,\left (A\,b^7-2\,B\,a\,b^6\right )\,{\left (a+b\,x\right )}^{11/2}}{768\,a^3}-\frac {5\,\left (A\,b^7-2\,B\,a\,b^6\right )\,{\left (a+b\,x\right )}^{13/2}}{1024\,a^4}+\frac {A\,b^7\,{\left (a+b\,x\right )}^{7/2}}{7\,a}}{7\,a\,{\left (a+b\,x\right )}^6-7\,a^6\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^7-21\,a^2\,{\left (a+b\,x\right )}^5+35\,a^3\,{\left (a+b\,x\right )}^4-35\,a^4\,{\left (a+b\,x\right )}^3+21\,a^5\,{\left (a+b\,x\right )}^2+a^7}-\frac {5\,b^6\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (A\,b-2\,B\,a\right )}{1024\,a^{9/2}} \]

[In]

int(((A + B*x)*(a + b*x)^(5/2))/x^8,x)

[Out]

(((283*A*b^7)/3072 - (283*B*a*b^6)/1536)*(a + b*x)^(5/2) + ((5*A*a^2*b^7)/1024 - (5*B*a^3*b^6)/512)*(a + b*x)^
(1/2) + ((25*B*a^2*b^6)/384 - (25*A*a*b^7)/768)*(a + b*x)^(3/2) - (283*(A*b^7 - 2*B*a*b^6)*(a + b*x)^(9/2))/(3
072*a^2) + (25*(A*b^7 - 2*B*a*b^6)*(a + b*x)^(11/2))/(768*a^3) - (5*(A*b^7 - 2*B*a*b^6)*(a + b*x)^(13/2))/(102
4*a^4) + (A*b^7*(a + b*x)^(7/2))/(7*a))/(7*a*(a + b*x)^6 - 7*a^6*(a + b*x) - (a + b*x)^7 - 21*a^2*(a + b*x)^5
+ 35*a^3*(a + b*x)^4 - 35*a^4*(a + b*x)^3 + 21*a^5*(a + b*x)^2 + a^7) - (5*b^6*atanh((a + b*x)^(1/2)/a^(1/2))*
(A*b - 2*B*a))/(1024*a^(9/2))