3.969 \(\int (d x)^m (c x^2)^{5/2} (a+b x) \, dx\)

Optimal. Leaf size=65 \[ \frac{a c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x} \]

[Out]

(a*c^2*(d*x)^(6 + m)*Sqrt[c*x^2])/(d^6*(6 + m)*x) + (b*c^2*(d*x)^(7 + m)*Sqrt[c*x^2])/(d^7*(7 + m)*x)

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Rubi [A]  time = 0.0304476, antiderivative size = 65, 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} \[ \frac{a c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*c^2*(d*x)^(6 + m)*Sqrt[c*x^2])/(d^6*(6 + m)*x) + (b*c^2*(d*x)^(7 + m)*Sqrt[c*x^2])/(d^7*(7 + 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 )^{5/2} (a+b x) \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int x^5 (d x)^m (a+b x) \, dx}{x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int (d x)^{5+m} (a+b x) \, dx}{d^5 x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \left (a (d x)^{5+m}+\frac{b (d x)^{6+m}}{d}\right ) \, dx}{d^5 x}\\ &=\frac{a c^2 (d x)^{6+m} \sqrt{c x^2}}{d^6 (6+m) x}+\frac{b c^2 (d x)^{7+m} \sqrt{c x^2}}{d^7 (7+m) x}\\ \end{align*}

Mathematica [A]  time = 0.0273481, size = 38, normalized size = 0.58 \[ \frac{x \left (c x^2\right )^{5/2} (d x)^m (a (m+7)+b (m+6) x)}{(m+6) (m+7)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(d*x)^m*(c*x^2)^(5/2)*(a*(7 + m) + b*(6 + m)*x))/((6 + m)*(7 + m))

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Maple [A]  time = 0.004, size = 40, normalized size = 0.6 \begin{align*}{\frac{ \left ( bmx+am+6\,bx+7\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 7+m \right ) \left ( 6+m \right ) } \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(c*x^2)^(5/2)*(b*x+a),x)

[Out]

x*(b*m*x+a*m+6*b*x+7*a)*(d*x)^m*(c*x^2)^(5/2)/(7+m)/(6+m)

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Maxima [A]  time = 1.06589, size = 53, normalized size = 0.82 \begin{align*} \frac{b c^{\frac{5}{2}} d^{m} x^{7} x^{m}}{m + 7} + \frac{a c^{\frac{5}{2}} d^{m} x^{6} x^{m}}{m + 6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*c^(5/2)*d^m*x^7*x^m/(m + 7) + a*c^(5/2)*d^m*x^6*x^m/(m + 6)

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Fricas [A]  time = 1.3552, size = 123, normalized size = 1.89 \begin{align*} \frac{{\left ({\left (b c^{2} m + 6 \, b c^{2}\right )} x^{6} +{\left (a c^{2} m + 7 \, a c^{2}\right )} x^{5}\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{2} + 13 \, m + 42} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*c^2*m + 6*b*c^2)*x^6 + (a*c^2*m + 7*a*c^2)*x^5)*sqrt(c*x^2)*(d*x)^m/(m^2 + 13*m + 42)

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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)**(5/2)*(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)^(5/2)*(b*x+a),x, algorithm="giac")

[Out]

Exception raised: TypeError