∫ (bx+cx2)3∕2 _________________

Optimal. Leaf size=76 \[ \frac {2 b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}-2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right ) \]

2/3*(c*x^2+b*x)^(3/2)/x^(3/2)-2*b^(3/2)*arctanh((c*x^2+b*x)^(1/2)/b^(1/2)/x^(1/2))+2*b*(c*x^2+b*x)^(1/2)/x^(1/

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {678, 674, 213} \begin {gather*} -2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )+\frac {2 b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}} \end {gather*}

(2*b*Sqrt[b*x + c*x^2])/Sqrt[x] + (2*(b*x + c*x^2)^(3/2))/(3*x^(3/2)) - 2*b^(3/2)*ArcTanh[Sqrt[b*x + c*x^2]/(S

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

, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(

2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((

a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1))), Int[(d + e*x)^(m + 1)*

(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a

*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx &=\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+b \int \frac {\sqrt {b x+c x^2}}{x^{3/2}} \, dx\\ &=\frac {2 b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+b^2 \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx\\ &=\frac {2 b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )\\ &=\frac {2 b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}-2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 70, normalized size = 0.92 \begin {gather*} \frac {2 \sqrt {x} \sqrt {b+c x} \left (\sqrt {b+c x} (4 b+c x)-3 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{3 \sqrt {x (b+c x)}} \end {gather*}

(2*Sqrt[x]*Sqrt[b + c*x]*(Sqrt[b + c*x]*(4*b + c*x) - 3*b^(3/2)*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/(3*Sqrt[x*(b

________________________________________________________________________________________


method	result	size

default	\(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (3 b^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-c x \sqrt {c x +b}-4 b \sqrt {c x +b}\right )}{3 \sqrt {x}\, \sqrt {c x +b}}\)	\(61\)

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

-2/3*(x*(c*x+b))^(1/2)*(3*b^(3/2)*arctanh((c*x+b)^(1/2)/b^(1/2))-c*x*(c*x+b)^(1/2)-4*b*(c*x+b)^(1/2))/x^(1/2)/

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

b*integrate(sqrt(c*x + b)/x, x) + 2/3*(c*x + b)^(3/2)

________________________________________________________________________________________

Fricas [A]
time = 1.54, size = 129, normalized size = 1.70 \begin {gather*} \left [\frac {3 \, b^{\frac {3}{2}} x \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, \sqrt {c x^{2} + b x} {\left (c x + 4 \, b\right )} \sqrt {x}}{3 \, x}, \frac {2 \, {\left (3 \, \sqrt {-b} b x \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + \sqrt {c x^{2} + b x} {\left (c x + 4 \, b\right )} \sqrt {x}\right )}}{3 \, x}\right ] \end {gather*}

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

[1/3*(3*b^(3/2)*x*log(-(c*x^2 + 2*b*x - 2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) + 2*sqrt(c*x^2 + b*x)*(c*x +

4*b)*sqrt(x))/x, 2/3*(3*sqrt(-b)*b*x*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) + sqrt(c*x^2 + b*x)*(c*x + 4*

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{x^{\frac {5}{2}}}\, dx \end {gather*}

________________________________________________________________________________________

Giac [A]
time = 2.28, size = 77, normalized size = 1.01 \begin {gather*} \frac {2 \, b^{2} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + \frac {2}{3} \, {\left (c x + b\right )}^{\frac {3}{2}} + 2 \, \sqrt {c x + b} b - \frac {2 \, {\left (3 \, b^{2} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b} b^{\frac {3}{2}}\right )}}{3 \, \sqrt {-b}} \end {gather*}

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

2*b^2*arctan(sqrt(c*x + b)/sqrt(-b))/sqrt(-b) + 2/3*(c*x + b)^(3/2) + 2*sqrt(c*x + b)*b - 2/3*(3*b^2*arctan(sq

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{x^{5/2}} \,d x \end {gather*}

________________________________________________________________________________________

3.1.88 \(\int \frac {(b x+c x^2)^{3/2}}{x^{5/2}} \, dx\) [88]