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

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

[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)

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Rubi [A]  time = 0.0294383, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {15, 16, 43} \[ \frac{a c \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac{b c \sqrt{c x^2} (d x)^{m+5}}{d^5 (m+5) x} \]

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*}

Mathematica [A]  time = 0.0304621, size = 38, normalized size = 0.62 \[ \frac{x \left (c x^2\right )^{3/2} (d x)^m (a (m+5)+b (m+4) x)}{(m+4) (m+5)} \]

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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Maple [A]  time = 0.002, size = 40, normalized size = 0.7 \begin{align*}{\frac{ \left ( bmx+am+4\,bx+5\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 5+m \right ) \left ( 4+m \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

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)/(5+m)/(4+m)

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Maxima [A]  time = 1.05781, size = 53, normalized size = 0.87 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.37498, size = 111, normalized size = 1.82 \begin{align*} \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{align*}

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

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]

Exception raised: TypeError

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

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