\(\int \frac {(a+b x)^{3/2} (A+B x)}{x^7} \, dx\) [409]

Optimal result

Integrand size = 18, antiderivative size = 208 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^7} \, dx=\frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}-\frac {b^3 (7 A b-12 a B) \sqrt {a+b x}}{768 a^3 x^2}+\frac {b^4 (7 A b-12 a B) \sqrt {a+b x}}{512 a^4 x}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}-\frac {b^5 (7 A b-12 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{9/2}} \]

[Out]

1/60*(7*A*b-12*B*a)*(b*x+a)^(3/2)/a/x^5-1/6*A*(b*x+a)^(5/2)/a/x^6-1/512*b^5*(7*A*b-12*B*a)*arctanh((b*x+a)^(1/
2)/a^(1/2))/a^(9/2)+1/160*b*(7*A*b-12*B*a)*(b*x+a)^(1/2)/a/x^4+1/960*b^2*(7*A*b-12*B*a)*(b*x+a)^(1/2)/a^2/x^3-
1/768*b^3*(7*A*b-12*B*a)*(b*x+a)^(1/2)/a^3/x^2+1/512*b^4*(7*A*b-12*B*a)*(b*x+a)^(1/2)/a^4/x

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 43, 44, 65, 214} \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^7} \, dx=-\frac {b^5 (7 A b-12 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{9/2}}+\frac {b^4 \sqrt {a+b x} (7 A b-12 a B)}{512 a^4 x}-\frac {b^3 \sqrt {a+b x} (7 A b-12 a B)}{768 a^3 x^2}+\frac {b^2 \sqrt {a+b x} (7 A b-12 a B)}{960 a^2 x^3}+\frac {(a+b x)^{3/2} (7 A b-12 a B)}{60 a x^5}+\frac {b \sqrt {a+b x} (7 A b-12 a B)}{160 a x^4}-\frac {A (a+b x)^{5/2}}{6 a x^6} \]

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/x^7,x]

[Out]

(b*(7*A*b - 12*a*B)*Sqrt[a + b*x])/(160*a*x^4) + (b^2*(7*A*b - 12*a*B)*Sqrt[a + b*x])/(960*a^2*x^3) - (b^3*(7*
A*b - 12*a*B)*Sqrt[a + b*x])/(768*a^3*x^2) + (b^4*(7*A*b - 12*a*B)*Sqrt[a + b*x])/(512*a^4*x) + ((7*A*b - 12*a
*B)*(a + b*x)^(3/2))/(60*a*x^5) - (A*(a + b*x)^(5/2))/(6*a*x^6) - (b^5*(7*A*b - 12*a*B)*ArcTanh[Sqrt[a + b*x]/
Sqrt[a]])/(512*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)^{5/2}}{6 a x^6}+\frac {\left (-\frac {7 A b}{2}+6 a B\right ) \int \frac {(a+b x)^{3/2}}{x^6} \, dx}{6 a} \\ & = \frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}-\frac {(b (7 A b-12 a B)) \int \frac {\sqrt {a+b x}}{x^5} \, dx}{40 a} \\ & = \frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}-\frac {\left (b^2 (7 A b-12 a B)\right ) \int \frac {1}{x^4 \sqrt {a+b x}} \, dx}{320 a} \\ & = \frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}+\frac {\left (b^3 (7 A b-12 a B)\right ) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{384 a^2} \\ & = \frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}-\frac {b^3 (7 A b-12 a B) \sqrt {a+b x}}{768 a^3 x^2}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}-\frac {\left (b^4 (7 A b-12 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{512 a^3} \\ & = \frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}-\frac {b^3 (7 A b-12 a B) \sqrt {a+b x}}{768 a^3 x^2}+\frac {b^4 (7 A b-12 a B) \sqrt {a+b x}}{512 a^4 x}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}+\frac {\left (b^5 (7 A b-12 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{1024 a^4} \\ & = \frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}-\frac {b^3 (7 A b-12 a B) \sqrt {a+b x}}{768 a^3 x^2}+\frac {b^4 (7 A b-12 a B) \sqrt {a+b x}}{512 a^4 x}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}+\frac {\left (b^4 (7 A b-12 a B)\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{512 a^4} \\ & = \frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}-\frac {b^3 (7 A b-12 a B) \sqrt {a+b x}}{768 a^3 x^2}+\frac {b^4 (7 A b-12 a B) \sqrt {a+b x}}{512 a^4 x}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}-\frac {b^5 (7 A b-12 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^7} \, dx=-\frac {\sqrt {a+b x} \left (-105 A b^5 x^5+48 a^3 b^2 x^2 (A+2 B x)+256 a^5 (5 A+6 B x)-8 a^2 b^3 x^3 (7 A+15 B x)+10 a b^4 x^4 (7 A+18 B x)+64 a^4 b x (26 A+33 B x)\right )}{7680 a^4 x^6}+\frac {b^5 (-7 A b+12 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{9/2}} \]

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/x^7,x]

[Out]

-1/7680*(Sqrt[a + b*x]*(-105*A*b^5*x^5 + 48*a^3*b^2*x^2*(A + 2*B*x) + 256*a^5*(5*A + 6*B*x) - 8*a^2*b^3*x^3*(7
*A + 15*B*x) + 10*a*b^4*x^4*(7*A + 18*B*x) + 64*a^4*b*x*(26*A + 33*B*x)))/(a^4*x^6) + (b^5*(-7*A*b + 12*a*B)*A
rcTanh[Sqrt[a + b*x]/Sqrt[a]])/(512*a^(9/2))

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.65

method	result	size
pseudoelliptic	\(\frac {-\frac {7 x^{6} b^{5} \left (A b -\frac {12 B a}{7}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{512}+\frac {7 \left (\frac {32 \left (-6 B x -5 A \right ) a^{\frac {11}{2}}}{7}+x \left (-\frac {5 x^{3} b^{3} \left (\frac {18 B x}{7}+A \right ) a^{\frac {3}{2}}}{4}+b^{2} x^{2} \left (\frac {15 B x}{7}+A \right ) a^{\frac {5}{2}}-\frac {6 b x \left (2 B x +A \right ) a^{\frac {7}{2}}}{7}+\frac {8 \left (-33 B x -26 A \right ) a^{\frac {9}{2}}}{7}+\frac {15 A \sqrt {a}\, b^{4} x^{4}}{8}\right ) b \right ) \sqrt {b x +a}}{960}}{a^{\frac {9}{2}} x^{6}}\)	\(135\)
risch	\(-\frac {\sqrt {b x +a}\, \left (-105 A \,b^{5} x^{5}+180 B a \,b^{4} x^{5}+70 a A \,b^{4} x^{4}-120 B \,a^{2} b^{3} x^{4}-56 a^{2} A \,b^{3} x^{3}+96 B \,a^{3} b^{2} x^{3}+48 a^{3} A \,b^{2} x^{2}+2112 B \,a^{4} b \,x^{2}+1664 a^{4} A b x +1536 a^{5} B x +1280 a^{5} A \right )}{7680 x^{6} a^{4}}-\frac {b^{5} \left (7 A b -12 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{512 a^{\frac {9}{2}}}\)	\(155\)
derivativedivides	\(2 b^{5} \left (-\frac {-\frac {\left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1024 a^{4}}+\frac {17 \left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{3072 a^{3}}-\frac {33 \left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{2560 a^{2}}+\frac {\left (281 A b -116 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{2560 a}+\left (\frac {119 A b}{3072}-\frac {17 B a}{256}\right ) \left (b x +a \right )^{\frac {3}{2}}-\frac {a \left (7 A b -12 B a \right ) \sqrt {b x +a}}{1024}}{b^{6} x^{6}}-\frac {\left (7 A b -12 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {9}{2}}}\right )\)	\(162\)
default	\(2 b^{5} \left (-\frac {-\frac {\left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1024 a^{4}}+\frac {17 \left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{3072 a^{3}}-\frac {33 \left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{2560 a^{2}}+\frac {\left (281 A b -116 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{2560 a}+\left (\frac {119 A b}{3072}-\frac {17 B a}{256}\right ) \left (b x +a \right )^{\frac {3}{2}}-\frac {a \left (7 A b -12 B a \right ) \sqrt {b x +a}}{1024}}{b^{6} x^{6}}-\frac {\left (7 A b -12 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {9}{2}}}\right )\)	\(162\)

[In]

int((b*x+a)^(3/2)*(B*x+A)/x^7,x,method=_RETURNVERBOSE)

[Out]

7/960*(-15/8*x^6*b^5*(A*b-12/7*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))+(32/7*(-6*B*x-5*A)*a^(11/2)+x*(-5/4*x^3*b^3
*(18/7*B*x+A)*a^(3/2)+b^2*x^2*(15/7*B*x+A)*a^(5/2)-6/7*b*x*(2*B*x+A)*a^(7/2)+8/7*(-33*B*x-26*A)*a^(9/2)+15/8*A
*a^(1/2)*b^4*x^4)*b)*(b*x+a)^(1/2))/a^(9/2)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.70 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^7} \, dx=\left [-\frac {15 \, {\left (12 \, B a b^{5} - 7 \, A b^{6}\right )} \sqrt {a} x^{6} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (1280 \, A a^{6} + 15 \, {\left (12 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} x^{5} - 10 \, {\left (12 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{4} + 8 \, {\left (12 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{3} + 48 \, {\left (44 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + 13 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{15360 \, a^{5} x^{6}}, -\frac {15 \, {\left (12 \, B a b^{5} - 7 \, A b^{6}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (1280 \, A a^{6} + 15 \, {\left (12 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} x^{5} - 10 \, {\left (12 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{4} + 8 \, {\left (12 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{3} + 48 \, {\left (44 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + 13 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{7680 \, a^{5} x^{6}}\right ] \]

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/x^7,x, algorithm="fricas")

[Out]

[-1/15360*(15*(12*B*a*b^5 - 7*A*b^6)*sqrt(a)*x^6*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(1280*A*a^6
+ 15*(12*B*a^2*b^4 - 7*A*a*b^5)*x^5 - 10*(12*B*a^3*b^3 - 7*A*a^2*b^4)*x^4 + 8*(12*B*a^4*b^2 - 7*A*a^3*b^3)*x^3
+ 48*(44*B*a^5*b + A*a^4*b^2)*x^2 + 128*(12*B*a^6 + 13*A*a^5*b)*x)*sqrt(b*x + a))/(a^5*x^6), -1/7680*(15*(12*
B*a*b^5 - 7*A*b^6)*sqrt(-a)*x^6*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (1280*A*a^6 + 15*(12*B*a^2*b^4 - 7*A*a*b^5)
*x^5 - 10*(12*B*a^3*b^3 - 7*A*a^2*b^4)*x^4 + 8*(12*B*a^4*b^2 - 7*A*a^3*b^3)*x^3 + 48*(44*B*a^5*b + A*a^4*b^2)*
x^2 + 128*(12*B*a^6 + 13*A*a^5*b)*x)*sqrt(b*x + a))/(a^5*x^6)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^7} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**(3/2)*(B*x+A)/x**7,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^7} \, dx=-\frac {1}{15360} \, b^{6} {\left (\frac {2 \, {\left (15 \, {\left (12 \, B a - 7 \, A b\right )} {\left (b x + a\right )}^{\frac {11}{2}} - 85 \, {\left (12 \, B a^{2} - 7 \, A a b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 198 \, {\left (12 \, B a^{3} - 7 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 6 \, {\left (116 \, B a^{4} - 281 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 85 \, {\left (12 \, B a^{5} - 7 \, A a^{4} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (12 \, B a^{6} - 7 \, A a^{5} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{6} a^{4} b - 6 \, {\left (b x + a\right )}^{5} a^{5} b + 15 \, {\left (b x + a\right )}^{4} a^{6} b - 20 \, {\left (b x + a\right )}^{3} a^{7} b + 15 \, {\left (b x + a\right )}^{2} a^{8} b - 6 \, {\left (b x + a\right )} a^{9} b + a^{10} b} + \frac {15 \, {\left (12 \, B a - 7 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}} b}\right )} \]

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/x^7,x, algorithm="maxima")

[Out]

-1/15360*b^6*(2*(15*(12*B*a - 7*A*b)*(b*x + a)^(11/2) - 85*(12*B*a^2 - 7*A*a*b)*(b*x + a)^(9/2) + 198*(12*B*a^
3 - 7*A*a^2*b)*(b*x + a)^(7/2) - 6*(116*B*a^4 - 281*A*a^3*b)*(b*x + a)^(5/2) - 85*(12*B*a^5 - 7*A*a^4*b)*(b*x
+ a)^(3/2) + 15*(12*B*a^6 - 7*A*a^5*b)*sqrt(b*x + a))/((b*x + a)^6*a^4*b - 6*(b*x + a)^5*a^5*b + 15*(b*x + a)^
4*a^6*b - 20*(b*x + a)^3*a^7*b + 15*(b*x + a)^2*a^8*b - 6*(b*x + a)*a^9*b + a^10*b) + 15*(12*B*a - 7*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 = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^7} \, dx=-\frac {\frac {15 \, {\left (12 \, B a b^{6} - 7 \, A b^{7}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {180 \, {\left (b x + a\right )}^{\frac {11}{2}} B a b^{6} - 1020 \, {\left (b x + a\right )}^{\frac {9}{2}} B a^{2} b^{6} + 2376 \, {\left (b x + a\right )}^{\frac {7}{2}} B a^{3} b^{6} - 696 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{4} b^{6} - 1020 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{5} b^{6} + 180 \, \sqrt {b x + a} B a^{6} b^{6} - 105 \, {\left (b x + a\right )}^{\frac {11}{2}} A b^{7} + 595 \, {\left (b x + a\right )}^{\frac {9}{2}} A a b^{7} - 1386 \, {\left (b x + a\right )}^{\frac {7}{2}} A a^{2} b^{7} + 1686 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{3} b^{7} + 595 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{4} b^{7} - 105 \, \sqrt {b x + a} A a^{5} b^{7}}{a^{4} b^{6} x^{6}}}{7680 \, b} \]

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/x^7,x, algorithm="giac")

[Out]

-1/7680*(15*(12*B*a*b^6 - 7*A*b^7)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + (180*(b*x + a)^(11/2)*B*a*b
^6 - 1020*(b*x + a)^(9/2)*B*a^2*b^6 + 2376*(b*x + a)^(7/2)*B*a^3*b^6 - 696*(b*x + a)^(5/2)*B*a^4*b^6 - 1020*(b
*x + a)^(3/2)*B*a^5*b^6 + 180*sqrt(b*x + a)*B*a^6*b^6 - 105*(b*x + a)^(11/2)*A*b^7 + 595*(b*x + a)^(9/2)*A*a*b
^7 - 1386*(b*x + a)^(7/2)*A*a^2*b^7 + 1686*(b*x + a)^(5/2)*A*a^3*b^7 + 595*(b*x + a)^(3/2)*A*a^4*b^7 - 105*sqr
t(b*x + a)*A*a^5*b^7)/(a^4*b^6*x^6))/b

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.22 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^7} \, dx=-\frac {\left (\frac {119\,A\,b^6}{1536}-\frac {17\,B\,a\,b^5}{128}\right )\,{\left (a+b\,x\right )}^{3/2}+\left (\frac {3\,B\,a^2\,b^5}{128}-\frac {7\,A\,a\,b^6}{512}\right )\,\sqrt {a+b\,x}-\frac {33\,\left (7\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{7/2}}{1280\,a^2}+\frac {17\,\left (7\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{9/2}}{1536\,a^3}-\frac {\left (7\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{11/2}}{512\,a^4}+\frac {\left (281\,A\,b^6-116\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{5/2}}{1280\,a}}{{\left (a+b\,x\right )}^6-6\,a^5\,\left (a+b\,x\right )-6\,a\,{\left (a+b\,x\right )}^5+15\,a^2\,{\left (a+b\,x\right )}^4-20\,a^3\,{\left (a+b\,x\right )}^3+15\,a^4\,{\left (a+b\,x\right )}^2+a^6}-\frac {b^5\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (7\,A\,b-12\,B\,a\right )}{512\,a^{9/2}} \]

[In]

int(((A + B*x)*(a + b*x)^(3/2))/x^7,x)

[Out]

- (((119*A*b^6)/1536 - (17*B*a*b^5)/128)*(a + b*x)^(3/2) + ((3*B*a^2*b^5)/128 - (7*A*a*b^6)/512)*(a + b*x)^(1/
2) - (33*(7*A*b^6 - 12*B*a*b^5)*(a + b*x)^(7/2))/(1280*a^2) + (17*(7*A*b^6 - 12*B*a*b^5)*(a + b*x)^(9/2))/(153
6*a^3) - ((7*A*b^6 - 12*B*a*b^5)*(a + b*x)^(11/2))/(512*a^4) + ((281*A*b^6 - 116*B*a*b^5)*(a + b*x)^(5/2))/(12
80*a))/((a + b*x)^6 - 6*a^5*(a + b*x) - 6*a*(a + b*x)^5 + 15*a^2*(a + b*x)^4 - 20*a^3*(a + b*x)^3 + 15*a^4*(a
+ b*x)^2 + a^6) - (b^5*atanh((a + b*x)^(1/2)/a^(1/2))*(7*A*b - 12*B*a))/(512*a^(9/2))