3.1.96 \(\int \frac {x^{3/2}}{\sqrt {b x+c x^2}} \, dx\) [96]

Optimal. Leaf size=52 \[ -\frac {4 b \sqrt {b x+c x^2}}{3 c^2 \sqrt {x}}+\frac {2 \sqrt {x} \sqrt {b x+c x^2}}{3 c} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {670, 662} \begin {gather*} \frac {2 \sqrt {x} \sqrt {b x+c x^2}}{3 c}-\frac {4 b \sqrt {b x+c x^2}}{3 c^2 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/Sqrt[b*x + c*x^2],x]

[Out]

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

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rubi steps

\begin {align*} \int \frac {x^{3/2}}{\sqrt {b x+c x^2}} \, dx &=\frac {2 \sqrt {x} \sqrt {b x+c x^2}}{3 c}-\frac {(2 b) \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx}{3 c}\\ &=-\frac {4 b \sqrt {b x+c x^2}}{3 c^2 \sqrt {x}}+\frac {2 \sqrt {x} \sqrt {b x+c x^2}}{3 c}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.58 \begin {gather*} \frac {2 (-2 b+c x) \sqrt {x (b+c x)}}{3 c^2 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/Sqrt[b*x + c*x^2],x]

[Out]

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

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Maple [A]
time = 0.41, size = 26, normalized size = 0.50

method	result	size
default	\(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (-c x +2 b \right )}{3 \sqrt {x}\, c^{2}}\)	\(26\)
risch	\(-\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (-c x +2 b \right )}{3 \sqrt {x \left (c x +b \right )}\, c^{2}}\)	\(31\)
gosper	\(-\frac {2 \left (c x +b \right ) \left (-c x +2 b \right ) \sqrt {x}}{3 c^{2} \sqrt {c \,x^{2}+b x}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(c*x^2+b*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 30, normalized size = 0.58 \begin {gather*} \frac {2 \, {\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )}}{3 \, \sqrt {c x + b} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.01, size = 26, normalized size = 0.50 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x} {\left (c x - 2 \, b\right )}}{3 \, c^{2} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)/(c*x**2+b*x)**(1/2),x)

[Out]

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

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Giac [A]
time = 1.05, size = 34, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}}}{3 \, c^{2}} - \frac {2 \, \sqrt {c x + b} b}{c^{2}} + \frac {4 \, b^{\frac {3}{2}}}{3 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{3/2}}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(b*x + c*x^2)^(1/2),x)

[Out]

int(x^(3/2)/(b*x + c*x^2)^(1/2), x)

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