3.1.73 \(\int x^{5/2} \sqrt {b x+c x^2} \, dx\)

Optimal. Leaf size=108 \[ -\frac {32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {656, 648} \begin {gather*} -\frac {32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*Sqrt[b*x + c*x^2],x]

[Out]

(-32*b^3*(b*x + c*x^2)^(3/2))/(315*c^4*x^(3/2)) + (16*b^2*(b*x + c*x^2)^(3/2))/(105*c^3*Sqrt[x]) - (4*b*Sqrt[x
]*(b*x + c*x^2)^(3/2))/(21*c^2) + (2*x^(3/2)*(b*x + c*x^2)^(3/2))/(9*c)

Rule 648

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 656

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 x^{5/2} \sqrt {b x+c x^2} \, dx &=\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac {(2 b) \int x^{3/2} \sqrt {b x+c x^2} \, dx}{3 c}\\ &=-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}+\frac {\left (8 b^2\right ) \int \sqrt {x} \sqrt {b x+c x^2} \, dx}{21 c^2}\\ &=\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac {\left (16 b^3\right ) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{105 c^3}\\ &=-\frac {32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 53, normalized size = 0.49 \begin {gather*} \frac {2 (x (b+c x))^{3/2} \left (-16 b^3+24 b^2 c x-30 b c^2 x^2+35 c^3 x^3\right )}{315 c^4 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*Sqrt[b*x + c*x^2],x]

[Out]

(2*(x*(b + c*x))^(3/2)*(-16*b^3 + 24*b^2*c*x - 30*b*c^2*x^2 + 35*c^3*x^3))/(315*c^4*x^(3/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.09, size = 66, normalized size = 0.61 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-16 b^4+8 b^3 c x-6 b^2 c^2 x^2+5 b c^3 x^3+35 c^4 x^4\right )}{315 c^4 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(5/2)*Sqrt[b*x + c*x^2],x]

[Out]

(2*Sqrt[b*x + c*x^2]*(-16*b^4 + 8*b^3*c*x - 6*b^2*c^2*x^2 + 5*b*c^3*x^3 + 35*c^4*x^4))/(315*c^4*Sqrt[x])

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 60, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt {c x^{2} + b x}}{315 \, c^{4} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*c^4*x^4 + 5*b*c^3*x^3 - 6*b^2*c^2*x^2 + 8*b^3*c*x - 16*b^4)*sqrt(c*x^2 + b*x)/(c^4*sqrt(x))

________________________________________________________________________________________

giac [A]  time = 0.17, size = 58, normalized size = 0.54 \begin {gather*} \frac {32 \, b^{\frac {9}{2}}}{315 \, c^{4}} + \frac {2 \, {\left (35 \, {\left (c x + b\right )}^{\frac {9}{2}} - 135 \, {\left (c x + b\right )}^{\frac {7}{2}} b + 189 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{2} - 105 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{3}\right )}}{315 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

32/315*b^(9/2)/c^4 + 2/315*(35*(c*x + b)^(9/2) - 135*(c*x + b)^(7/2)*b + 189*(c*x + b)^(5/2)*b^2 - 105*(c*x +
b)^(3/2)*b^3)/c^4

________________________________________________________________________________________

maple [A]  time = 0.05, size = 55, normalized size = 0.51 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-35 x^{3} c^{3}+30 b \,x^{2} c^{2}-24 b^{2} x c +16 b^{3}\right ) \sqrt {c \,x^{2}+b x}}{315 c^{4} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(c*x^2+b*x)^(1/2),x)

[Out]

-2/315*(c*x+b)*(-35*c^3*x^3+30*b*c^2*x^2-24*b^2*c*x+16*b^3)*(c*x^2+b*x)^(1/2)/c^4/x^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.49, size = 53, normalized size = 0.49 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt {c x + b}}{315 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

2/315*(35*c^4*x^4 + 5*b*c^3*x^3 - 6*b^2*c^2*x^2 + 8*b^3*c*x - 16*b^4)*sqrt(c*x + b)/c^4

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^{5/2}\,\sqrt {c\,x^2+b\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x + c*x^2)^(1/2),x)

[Out]

int(x^(5/2)*(b*x + c*x^2)^(1/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(c*x**2+b*x)**(1/2),x)

[Out]

Integral(x**(5/2)*sqrt(x*(b + c*x)), x)

________________________________________________________________________________________