\(\int \frac {(a+b x)^{5/2} (A+B x)}{x^2} \, dx\) [416]

Optimal result

Integrand size = 18, antiderivative size = 118 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^2} \, dx=a (5 A b+2 a B) \sqrt {a+b x}+\frac {1}{3} (5 A b+2 a B) (a+b x)^{3/2}+\frac {(5 A b+2 a B) (a+b x)^{5/2}}{5 a}-\frac {A (a+b x)^{7/2}}{a x}-a^{3/2} (5 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]

[Out]

1/3*(5*A*b+2*B*a)*(b*x+a)^(3/2)+1/5*(5*A*b+2*B*a)*(b*x+a)^(5/2)/a-A*(b*x+a)^(7/2)/a/x-a^(3/2)*(5*A*b+2*B*a)*ar
ctanh((b*x+a)^(1/2)/a^(1/2))+a*(5*A*b+2*B*a)*(b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 52, 65, 214} \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^2} \, dx=-a^{3/2} (2 a B+5 A b) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {(a+b x)^{5/2} (2 a B+5 A b)}{5 a}+\frac {1}{3} (a+b x)^{3/2} (2 a B+5 A b)+a \sqrt {a+b x} (2 a B+5 A b)-\frac {A (a+b x)^{7/2}}{a x} \]

[In]

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

[Out]

a*(5*A*b + 2*a*B)*Sqrt[a + b*x] + ((5*A*b + 2*a*B)*(a + b*x)^(3/2))/3 + ((5*A*b + 2*a*B)*(a + b*x)^(5/2))/(5*a
) - (A*(a + b*x)^(7/2))/(a*x) - a^(3/2)*(5*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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}}{a x}+\frac {\left (\frac {5 A b}{2}+a B\right ) \int \frac {(a+b x)^{5/2}}{x} \, dx}{a} \\ & = \frac {(5 A b+2 a B) (a+b x)^{5/2}}{5 a}-\frac {A (a+b x)^{7/2}}{a x}+\frac {1}{2} (5 A b+2 a B) \int \frac {(a+b x)^{3/2}}{x} \, dx \\ & = \frac {1}{3} (5 A b+2 a B) (a+b x)^{3/2}+\frac {(5 A b+2 a B) (a+b x)^{5/2}}{5 a}-\frac {A (a+b x)^{7/2}}{a x}+\frac {1}{2} (a (5 A b+2 a B)) \int \frac {\sqrt {a+b x}}{x} \, dx \\ & = a (5 A b+2 a B) \sqrt {a+b x}+\frac {1}{3} (5 A b+2 a B) (a+b x)^{3/2}+\frac {(5 A b+2 a B) (a+b x)^{5/2}}{5 a}-\frac {A (a+b x)^{7/2}}{a x}+\frac {1}{2} \left (a^2 (5 A b+2 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx \\ & = a (5 A b+2 a B) \sqrt {a+b x}+\frac {1}{3} (5 A b+2 a B) (a+b x)^{3/2}+\frac {(5 A b+2 a B) (a+b x)^{5/2}}{5 a}-\frac {A (a+b x)^{7/2}}{a x}+\frac {\left (a^2 (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 )}{b} \\ & = a (5 A b+2 a B) \sqrt {a+b x}+\frac {1}{3} (5 A b+2 a B) (a+b x)^{3/2}+\frac {(5 A b+2 a B) (a+b x)^{5/2}}{5 a}-\frac {A (a+b x)^{7/2}}{a x}-a^{3/2} (5 A b+2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^2} \, dx=\frac {\sqrt {a+b x} \left (2 b^2 x^2 (5 A+3 B x)+2 a b x (35 A+11 B x)+a^2 (-15 A+46 B x)\right )}{15 x}-a^{3/2} (5 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]

[In]

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

[Out]

(Sqrt[a + b*x]*(2*b^2*x^2*(5*A + 3*B*x) + 2*a*b*x*(35*A + 11*B*x) + a^2*(-15*A + 46*B*x)))/(15*x) - a^(3/2)*(5
*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(\frac {\frac {2 \left (-\frac {15}{2} a^{2} b A -3 a^{3} B \right ) x \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{3}+\frac {2 \sqrt {b x +a}\, \left (7 x \left (\frac {11 B x}{35}+A \right ) b \,a^{\frac {3}{2}}+\left (\frac {23 B x}{5}-\frac {3 A}{2}\right ) a^{\frac {5}{2}}+b^{2} x^{2} \sqrt {a}\, \left (\frac {3 B x}{5}+A \right )\right )}{3}}{x \sqrt {a}}\)	\(88\)
risch	\(-\frac {a^{2} A \sqrt {b x +a}}{x}+\frac {2 B \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 A b \left (b x +a \right )^{\frac {3}{2}}}{3}+\frac {2 B a \left (b x +a \right )^{\frac {3}{2}}}{3}+4 A a b \sqrt {b x +a}+2 B \,a^{2} \sqrt {b x +a}-a^{\frac {3}{2}} \left (5 A b +2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\)	\(101\)
derivativedivides	\(\frac {2 B \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 A b \left (b x +a \right )^{\frac {3}{2}}}{3}+\frac {2 B a \left (b x +a \right )^{\frac {3}{2}}}{3}+4 A a b \sqrt {b x +a}+2 B \,a^{2} \sqrt {b x +a}-2 a^{2} \left (\frac {A \sqrt {b x +a}}{2 x}+\frac {\left (5 A b +2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\)	\(104\)
default	\(\frac {2 B \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 A b \left (b x +a \right )^{\frac {3}{2}}}{3}+\frac {2 B a \left (b x +a \right )^{\frac {3}{2}}}{3}+4 A a b \sqrt {b x +a}+2 B \,a^{2} \sqrt {b x +a}-2 a^{2} \left (\frac {A \sqrt {b x +a}}{2 x}+\frac {\left (5 A b +2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\)	\(104\)

[In]

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

[Out]

2/3/a^(1/2)*((-15/2*a^2*b*A-3*a^3*B)*x*arctanh((b*x+a)^(1/2)/a^(1/2))+(b*x+a)^(1/2)*(7*x*(11/35*B*x+A)*b*a^(3/
2)+(23/5*B*x-3/2*A)*a^(5/2)+b^2*x^2*a^(1/2)*(3/5*B*x+A)))/x

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^2} \, dx=\left [\frac {15 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} \sqrt {a} x \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (6 \, B b^{2} x^{3} - 15 \, A a^{2} + 2 \, {\left (11 \, B a b + 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (23 \, B a^{2} + 35 \, A a b\right )} x\right )} \sqrt {b x + a}}{30 \, x}, \frac {15 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (6 \, B b^{2} x^{3} - 15 \, A a^{2} + 2 \, {\left (11 \, B a b + 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (23 \, B a^{2} + 35 \, A a b\right )} x\right )} \sqrt {b x + a}}{15 \, x}\right ] \]

[In]

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

[Out]

[1/30*(15*(2*B*a^2 + 5*A*a*b)*sqrt(a)*x*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(6*B*b^2*x^3 - 15*A*a
^2 + 2*(11*B*a*b + 5*A*b^2)*x^2 + 2*(23*B*a^2 + 35*A*a*b)*x)*sqrt(b*x + a))/x, 1/15*(15*(2*B*a^2 + 5*A*a*b)*sq
rt(-a)*x*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (6*B*b^2*x^3 - 15*A*a^2 + 2*(11*B*a*b + 5*A*b^2)*x^2 + 2*(23*B*a^2
+ 35*A*a*b)*x)*sqrt(b*x + a))/x]

Sympy [A] (verification not implemented)

Time = 12.39 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.14 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^2} \, dx=- A a^{\frac {3}{2}} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} - \frac {A a^{2} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{\sqrt {x}} + 2 A a b \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + b x} & \text {for}\: b \neq 0 \\\sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + A b^{2} \left (\begin {cases} \frac {2 \left (a + b x\right )^{\frac {3}{2}}}{3 b} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) + B a^{2} \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + b x} & \text {for}\: b \neq 0 \\\sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + 2 B a b \left (\begin {cases} \frac {2 \left (a + b x\right )^{\frac {3}{2}}}{3 b} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) + B b^{2} \left (\begin {cases} - \frac {2 a \left (a + b x\right )^{\frac {3}{2}}}{3 b^{2}} + \frac {2 \left (a + b x\right )^{\frac {5}{2}}}{5 b^{2}} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{2}}{2} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

-A*a**(3/2)*b*asinh(sqrt(a)/(sqrt(b)*sqrt(x))) - A*a**2*sqrt(b)*sqrt(a/(b*x) + 1)/sqrt(x) + 2*A*a*b*Piecewise(
(2*a*atan(sqrt(a + b*x)/sqrt(-a))/sqrt(-a) + 2*sqrt(a + b*x), Ne(b, 0)), (sqrt(a)*log(x), True)) + A*b**2*Piec
ewise((2*(a + b*x)**(3/2)/(3*b), Ne(b, 0)), (sqrt(a)*x, True)) + B*a**2*Piecewise((2*a*atan(sqrt(a + b*x)/sqrt
(-a))/sqrt(-a) + 2*sqrt(a + b*x), Ne(b, 0)), (sqrt(a)*log(x), True)) + 2*B*a*b*Piecewise((2*(a + b*x)**(3/2)/(
3*b), Ne(b, 0)), (sqrt(a)*x, True)) + B*b**2*Piecewise((-2*a*(a + b*x)**(3/2)/(3*b**2) + 2*(a + b*x)**(5/2)/(5
*b**2), Ne(b, 0)), (sqrt(a)*x**2/2, True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^2} \, dx=\frac {1}{30} \, {\left (\frac {15 \, {\left (2 \, B a + 5 \, A b\right )} a^{\frac {3}{2}} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{b} - \frac {30 \, \sqrt {b x + a} A a^{2}}{b x} + \frac {4 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} B + 5 \, {\left (B a + A b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} \sqrt {b x + a}\right )}}{b}\right )} b \]

[In]

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

[Out]

1/30*(15*(2*B*a + 5*A*b)*a^(3/2)*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/b - 30*sqrt(b*x + a)
*A*a^2/(b*x) + 4*(3*(b*x + a)^(5/2)*B + 5*(B*a + A*b)*(b*x + a)^(3/2) + 15*(B*a^2 + 2*A*a*b)*sqrt(b*x + a))/b)
*b

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^2} \, dx=\frac {6 \, {\left (b x + a\right )}^{\frac {5}{2}} B b + 10 \, {\left (b x + a\right )}^{\frac {3}{2}} B a b + 30 \, \sqrt {b x + a} B a^{2} b + 10 \, {\left (b x + a\right )}^{\frac {3}{2}} A b^{2} + 60 \, \sqrt {b x + a} A a b^{2} - \frac {15 \, \sqrt {b x + a} A a^{2} b}{x} + \frac {15 \, {\left (2 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}}}{15 \, b} \]

[In]

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

[Out]

1/15*(6*(b*x + a)^(5/2)*B*b + 10*(b*x + a)^(3/2)*B*a*b + 30*sqrt(b*x + a)*B*a^2*b + 10*(b*x + a)^(3/2)*A*b^2 +
60*sqrt(b*x + a)*A*a*b^2 - 15*sqrt(b*x + a)*A*a^2*b/x + 15*(2*B*a^3*b + 5*A*a^2*b^2)*arctan(sqrt(b*x + a)/sqr
t(-a))/sqrt(-a))/b

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^2} \, dx=\left (\frac {2\,A\,b}{3}+\frac {2\,B\,a}{3}\right )\,{\left (a+b\,x\right )}^{3/2}+\left (2\,a\,\left (2\,A\,b+2\,B\,a\right )-2\,B\,a^2\right )\,\sqrt {a+b\,x}+\frac {2\,B\,{\left (a+b\,x\right )}^{5/2}}{5}-\frac {A\,a^2\,\sqrt {a+b\,x}}{x}+a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\left (5\,A\,b+2\,B\,a\right )\,1{}\mathrm {i} \]

[In]

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

[Out]

((2*A*b)/3 + (2*B*a)/3)*(a + b*x)^(3/2) + (2*a*(2*A*b + 2*B*a) - 2*B*a^2)*(a + b*x)^(1/2) + (2*B*(a + b*x)^(5/
2))/5 + a^(3/2)*atan(((a + b*x)^(1/2)*1i)/a^(1/2))*(5*A*b + 2*B*a)*1i - (A*a^2*(a + b*x)^(1/2))/x