3.8.25 \(\int (c x^2)^{3/2} (a+b x) \, dx\)

Optimal. Leaf size=37 \[ \frac {1}{4} a c x^3 \sqrt {c x^2}+\frac {1}{5} b c x^4 \sqrt {c x^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 43} \begin {gather*} \frac {1}{4} a c x^3 \sqrt {c x^2}+\frac {1}{5} b c x^4 \sqrt {c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(3/2)*(a + b*x),x]

[Out]

(a*c*x^3*Sqrt[c*x^2])/4 + (b*c*x^4*Sqrt[c*x^2])/5

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \left (c x^2\right )^{3/2} (a+b x) \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x^3 (a+b x) \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (a x^3+b x^4\right ) \, dx}{x}\\ &=\frac {1}{4} a c x^3 \sqrt {c x^2}+\frac {1}{5} b c x^4 \sqrt {c x^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.59 \begin {gather*} \frac {1}{20} x \left (c x^2\right )^{3/2} (5 a+4 b x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*x^2)^(3/2)*(a + b*x),x]

[Out]

(x*(c*x^2)^(3/2)*(5*a + 4*b*x))/20

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IntegrateAlgebraic [A]  time = 0.02, size = 22, normalized size = 0.59 \begin {gather*} \frac {1}{20} x \left (c x^2\right )^{3/2} (5 a+4 b x) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c*x^2)^(3/2)*(a + b*x),x]

[Out]

(x*(c*x^2)^(3/2)*(5*a + 4*b*x))/20

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fricas [A]  time = 0.83, size = 24, normalized size = 0.65 \begin {gather*} \frac {1}{20} \, {\left (4 \, b c x^{4} + 5 \, a c x^{3}\right )} \sqrt {c x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*b*c*x^4 + 5*a*c*x^3)*sqrt(c*x^2)

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giac [A]  time = 0.88, size = 22, normalized size = 0.59 \begin {gather*} \frac {1}{20} \, {\left (4 \, b x^{5} \mathrm {sgn}\relax (x) + 5 \, a x^{4} \mathrm {sgn}\relax (x)\right )} c^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*b*x^5*sgn(x) + 5*a*x^4*sgn(x))*c^(3/2)

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maple [A]  time = 0.00, size = 19, normalized size = 0.51 \begin {gather*} \frac {\left (4 b x +5 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}} x}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)*(b*x+a),x)

[Out]

1/20*x*(4*b*x+5*a)*(c*x^2)^(3/2)

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maxima [A]  time = 1.30, size = 25, normalized size = 0.68 \begin {gather*} \frac {1}{4} \, \left (c x^{2}\right )^{\frac {3}{2}} a x + \frac {\left (c x^{2}\right )^{\frac {5}{2}} b}{5 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c*x^2)^(3/2)*a*x + 1/5*(c*x^2)^(5/2)*b/c

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)*(a + b*x),x)

[Out]

int((c*x^2)^(3/2)*(a + b*x), x)

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sympy [A]  time = 0.56, size = 34, normalized size = 0.92 \begin {gather*} \frac {a c^{\frac {3}{2}} x \left (x^{2}\right )^{\frac {3}{2}}}{4} + \frac {b c^{\frac {3}{2}} x^{2} \left (x^{2}\right )^{\frac {3}{2}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**(3/2)*x*(x**2)**(3/2)/4 + b*c**(3/2)*x**2*(x**2)**(3/2)/5

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