\(\int \frac {x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx\) [539]

Optimal result

Integrand size = 20, antiderivative size = 133 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt {a+b x}}-\frac {(2 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{a b^3}+\frac {(2 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{7/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {79, 49, 52, 65, 223, 212} \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {(2 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{7/2}}-\frac {\sqrt {x} \sqrt {a+b x} (2 A b-5 a B)}{a b^3}+\frac {2 x^{3/2} (2 A b-5 a B)}{3 a b^2 \sqrt {a+b x}}+\frac {2 x^{5/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

[In]

Int[(x^(3/2)*(A + B*x))/(a + b*x)^(5/2),x]

[Out]

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

Rule 49

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.77 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {x} \left (-6 a A b+15 a^2 B-8 A b^2 x+20 a b B x+3 b^2 B x^2\right )}{3 b^3 (a+b x)^{3/2}}+\frac {2 (2 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{b^{7/2}} \]

[In]

Integrate[(x^(3/2)*(A + B*x))/(a + b*x)^(5/2),x]

[Out]

(Sqrt[x]*(-6*a*A*b + 15*a^2*B - 8*A*b^2*x + 20*a*b*B*x + 3*b^2*B*x^2))/(3*b^3*(a + b*x)^(3/2)) + (2*(2*A*b - 5
*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])])/b^(7/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(243\) vs. \(2(109)=218\).

Time = 0.54 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.83

method	result	size
risch	\(\frac {B \sqrt {x}\, \sqrt {b x +a}}{b^{3}}+\frac {\left (2 A \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )-\frac {5 B a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {b}}-\frac {4 \left (2 A b -3 B a \right ) \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{b \left (x +\frac {a}{b}\right )}+\frac {2 a^{2} \left (A b -B a \right ) \left (\frac {2 \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{3 a \left (x +\frac {a}{b}\right )^{2}}+\frac {4 b \sqrt {b \left (x +\frac {a}{b}\right )^{2}-\left (x +\frac {a}{b}\right ) a}}{3 a^{2} \left (x +\frac {a}{b}\right )}\right )}{b^{2}}\right ) \sqrt {x \left (b x +a \right )}}{2 b^{3} \sqrt {x}\, \sqrt {b x +a}}\)	\(244\)
default	\(\frac {\left (6 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) b^{3} x^{2}-15 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a \,b^{2} x^{2}+6 B \,b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+12 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a \,b^{2} x -16 A \,b^{\frac {5}{2}} \sqrt {x \left (b x +a \right )}\, x -30 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} b x +40 B \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a x +6 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} b -12 A \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a -15 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3}+30 B \sqrt {b}\, \sqrt {x \left (b x +a \right )}\, a^{2}\right ) \sqrt {x}}{6 \sqrt {x \left (b x +a \right )}\, b^{\frac {7}{2}} \left (b x +a \right )^{\frac {3}{2}}}\)	\(315\)

[In]

int(x^(3/2)*(B*x+A)/(b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

B/b^3*x^(1/2)*(b*x+a)^(1/2)+1/2/b^3*(2*A*b^(1/2)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))-5*B*a*ln((1/2*a+b*x
)/b^(1/2)+(b*x^2+a*x)^(1/2))/b^(1/2)-4*(2*A*b-3*B*a)/b/(x+a/b)*(b*(x+a/b)^2-(x+a/b)*a)^(1/2)+2*a^2*(A*b-B*a)/b
^2*(2/3/a/(x+a/b)^2*(b*(x+a/b)^2-(x+a/b)*a)^(1/2)+4/3*b/a^2/(x+a/b)*(b*(x+a/b)^2-(x+a/b)*a)^(1/2)))*(x*(b*x+a)
)^(1/2)/x^(1/2)/(b*x+a)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.36 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\left [-\frac {3 \, {\left (5 \, B a^{3} - 2 \, A a^{2} b + {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (3 \, B b^{3} x^{2} + 15 \, B a^{2} b - 6 \, A a b^{2} + 4 \, {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{6 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {3 \, {\left (5 \, B a^{3} - 2 \, A a^{2} b + {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (3 \, B b^{3} x^{2} + 15 \, B a^{2} b - 6 \, A a b^{2} + 4 \, {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]

[In]

integrate(x^(3/2)*(B*x+A)/(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(3*(5*B*a^3 - 2*A*a^2*b + (5*B*a*b^2 - 2*A*b^3)*x^2 + 2*(5*B*a^2*b - 2*A*a*b^2)*x)*sqrt(b)*log(2*b*x + 2
*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(3*B*b^3*x^2 + 15*B*a^2*b - 6*A*a*b^2 + 4*(5*B*a*b^2 - 2*A*b^3)*x)*sqr
t(b*x + a)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), 1/3*(3*(5*B*a^3 - 2*A*a^2*b + (5*B*a*b^2 - 2*A*b^3)*x^2 +
2*(5*B*a^2*b - 2*A*a*b^2)*x)*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (3*B*b^3*x^2 + 15*B*a^2*b
- 6*A*a*b^2 + 4*(5*B*a*b^2 - 2*A*b^3)*x)*sqrt(b*x + a)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (126) = 252\).

Time = 51.80 (sec) , antiderivative size = 729, normalized size of antiderivative = 5.48 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx=A \left (\frac {6 a^{\frac {39}{2}} b^{11} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}} + \frac {6 a^{\frac {37}{2}} b^{12} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}} - \frac {6 a^{19} b^{\frac {23}{2}} x^{14}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}} - \frac {8 a^{18} b^{\frac {25}{2}} x^{15}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}}\right ) + B \left (- \frac {15 a^{\frac {81}{2}} b^{22} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}}} - \frac {15 a^{\frac {79}{2}} b^{23} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}}} + \frac {15 a^{40} b^{\frac {45}{2}} x^{26}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}}} + \frac {20 a^{39} b^{\frac {47}{2}} x^{27}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}}} + \frac {3 a^{38} b^{\frac {49}{2}} x^{28}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}}}\right ) \]

[In]

integrate(x**(3/2)*(B*x+A)/(b*x+a)**(5/2),x)

[Out]

A*(6*a**(39/2)*b**11*x**(27/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)
*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a)) + 6*a**(37/2)*b**12*x**(29/2)*sqrt(1 + b*x
/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x*
*(29/2)*sqrt(1 + b*x/a)) - 6*a**19*b**(23/2)*x**14/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37
/2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a)) - 8*a**18*b**(25/2)*x**15/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b
*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a))) + B*(-15*a**(81/2)*b**22*x**(51/2)*sqrt(1 + b*x/a)*a
sinh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)*x**(53/
2)*sqrt(1 + b*x/a)) - 15*a**(79/2)*b**23*x**(53/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(79/2)
*b**(51/2)*x**(51/2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 + b*x/a)) + 15*a**40*b**(45/2)*x
**26/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 + b*x/a)) + 20*
a**39*b**(47/2)*x**27/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(
1 + b*x/a)) + 3*a**38*b**(49/2)*x**28/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)
*x**(53/2)*sqrt(1 + b*x/a)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (109) = 218\).

Time = 0.22 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.51 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{3 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} - \frac {\sqrt {b x^{2} + a x} B a^{2}}{3 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}} + \frac {\sqrt {b x^{2} + a x} A a}{3 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} + \frac {16 \, \sqrt {b x^{2} + a x} B a}{3 \, {\left (b^{4} x + a b^{3}\right )}} - \frac {7 \, \sqrt {b x^{2} + a x} A}{3 \, {\left (b^{3} x + a b^{2}\right )}} - \frac {5 \, B a \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{2 \, b^{\frac {7}{2}}} + \frac {A \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{b^{\frac {5}{2}}} \]

[In]

integrate(x^(3/2)*(B*x+A)/(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

1/3*(b*x^2 + a*x)^(3/2)*B*a/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3*b^2) - 1/3*sqrt(b*x^2 + a*x)*B*a^2/(b^5
*x^2 + 2*a*b^4*x + a^2*b^3) - 1/3*(b*x^2 + a*x)^(3/2)*A/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b) + (b*x^2
+ a*x)^(3/2)*B/(b^4*x^2 + 2*a*b^3*x + a^2*b^2) + 1/3*sqrt(b*x^2 + a*x)*A*a/(b^4*x^2 + 2*a*b^3*x + a^2*b^2) +
16/3*sqrt(b*x^2 + a*x)*B*a/(b^4*x + a*b^3) - 7/3*sqrt(b*x^2 + a*x)*A/(b^3*x + a*b^2) - 5/2*B*a*log(2*x + a/b +
2*sqrt(b*x^2 + a*x)/sqrt(b))/b^(7/2) + A*log(2*x + a/b + 2*sqrt(b*x^2 + a*x)/sqrt(b))/b^(5/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (109) = 218\).

Time = 16.04 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.23 \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} B {\left | b \right |}}{b^{5}} + \frac {{\left (5 \, B a {\left | b \right |} - 2 \, A b {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{2 \, b^{\frac {9}{2}}} + \frac {4 \, {\left (9 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} {\left | b \right |} + 12 \, B a^{3} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b {\left | b \right |} - 6 \, A a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b {\left | b \right |} + 7 \, B a^{4} b^{2} {\left | b \right |} - 6 \, A a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{2} {\left | b \right |} - 4 \, A a^{3} b^{3} {\left | b \right |}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{\frac {7}{2}}} \]

[In]

integrate(x^(3/2)*(B*x+A)/(b*x+a)^(5/2),x, algorithm="giac")

[Out]

sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*B*abs(b)/b^5 + 1/2*(5*B*a*abs(b) - 2*A*b*abs(b))*log((sqrt(b*x + a)*sqrt
(b) - sqrt((b*x + a)*b - a*b))^2)/b^(9/2) + 4/3*(9*B*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*a
bs(b) + 12*B*a^3*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b*abs(b) - 6*A*a*(sqrt(b*x + a)*sqrt(b) -
sqrt((b*x + a)*b - a*b))^4*b*abs(b) + 7*B*a^4*b^2*abs(b) - 6*A*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b
- a*b))^2*b^2*abs(b) - 4*A*a^3*b^3*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*b^(7
/2))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx=\int \frac {x^{3/2}\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^{5/2}} \,d x \]

[In]

int((x^(3/2)*(A + B*x))/(a + b*x)^(5/2),x)

[Out]

int((x^(3/2)*(A + B*x))/(a + b*x)^(5/2), x)