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

Optimal. Leaf size=61 \[ \frac {a c \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac {b c \sqrt {c x^2} (d x)^{m+5}}{d^5 (m+5) x} \]

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Rubi [A]  time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {15, 16, 43} \begin {gather*} \frac {a c \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac {b c \sqrt {c x^2} (d x)^{m+5}}{d^5 (m+5) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(c*x^2)^(3/2)*(a + b*x),x]

[Out]

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

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 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, 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 (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x^3 (d x)^m (a+b x) \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int (d x)^{3+m} (a+b x) \, dx}{d^3 x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (a (d x)^{3+m}+\frac {b (d x)^{4+m}}{d}\right ) \, dx}{d^3 x}\\ &=\frac {a c (d x)^{4+m} \sqrt {c x^2}}{d^4 (4+m) x}+\frac {b c (d x)^{5+m} \sqrt {c x^2}}{d^5 (5+m) x}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.62 \begin {gather*} \frac {x \left (c x^2\right )^{3/2} (d x)^m (a (m+5)+b (m+4) x)}{(m+4) (m+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(c*x^2)^(3/2)*(a + b*x),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.39, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (c x^2\right )^{3/2} (a+b x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.49, size = 50, normalized size = 0.82 \begin {gather*} \frac {{\left ({\left (b c m + 4 \, b c\right )} x^{4} + {\left (a c m + 5 \, a c\right )} x^{3}\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{2} + 9 \, m + 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*c*m + 4*b*c)*x^4 + (a*c*m + 5*a*c)*x^3)*sqrt(c*x^2)*(d*x)^m/(m^2 + 9*m + 20)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Undef/Unsigned Inf encountered in limit

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maple [A]  time = 0.00, size = 40, normalized size = 0.66 \begin {gather*} \frac {\left (b m x +a m +4 b x +5 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}} x \left (d x \right )^{m}}{\left (m +5\right ) \left (m +4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.55, size = 39, normalized size = 0.64 \begin {gather*} \frac {b c^{\frac {3}{2}} d^{m} x^{5} x^{m}}{m + 5} + \frac {a c^{\frac {3}{2}} d^{m} x^{4} x^{m}}{m + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.24, size = 42, normalized size = 0.69 \begin {gather*} \frac {c\,x^3\,{\left (d\,x\right )}^m\,\sqrt {c\,x^2}\,\left (5\,a+a\,m+4\,b\,x+b\,m\,x\right )}{m^2+9\,m+20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x^3*(d*x)^m*(c*x^2)^(1/2)*(5*a + a*m + 4*b*x + b*m*x))/(9*m + m^2 + 20)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {\int \frac {a \left (c x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx + \int \frac {b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx}{d^{5}} & \text {for}\: m = -5 \\\frac {\int \frac {a \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx + \int \frac {b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx}{d^{4}} & \text {for}\: m = -4 \\\frac {a c^{\frac {3}{2}} d^{m} m x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{2} + 9 m + 20} + \frac {5 a c^{\frac {3}{2}} d^{m} x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{2} + 9 m + 20} + \frac {b c^{\frac {3}{2}} d^{m} m x^{2} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{2} + 9 m + 20} + \frac {4 b c^{\frac {3}{2}} d^{m} x^{2} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{2} + 9 m + 20} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((Integral(a*(c*x**2)**(3/2)/x**5, x) + Integral(b*(c*x**2)**(3/2)/x**4, x))/d**5, Eq(m, -5)), ((Int
egral(a*(c*x**2)**(3/2)/x**4, x) + Integral(b*(c*x**2)**(3/2)/x**3, x))/d**4, Eq(m, -4)), (a*c**(3/2)*d**m*m*x
*x**m*(x**2)**(3/2)/(m**2 + 9*m + 20) + 5*a*c**(3/2)*d**m*x*x**m*(x**2)**(3/2)/(m**2 + 9*m + 20) + b*c**(3/2)*
d**m*m*x**2*x**m*(x**2)**(3/2)/(m**2 + 9*m + 20) + 4*b*c**(3/2)*d**m*x**2*x**m*(x**2)**(3/2)/(m**2 + 9*m + 20)
, True))

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