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

Optimal. Leaf size=67 \[ -\frac{a d^4 x (d x)^{m-4}}{c^2 (4-m) \sqrt{c x^2}}-\frac{b d^3 x (d x)^{m-3}}{c^2 (3-m) \sqrt{c x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0365748, antiderivative size = 67, 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} \[ -\frac{a d^4 x (d x)^{m-4}}{c^2 (4-m) \sqrt{c x^2}}-\frac{b d^3 x (d x)^{m-3}}{c^2 (3-m) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0547224, size = 38, normalized size = 0.57 \[ \frac{x (d x)^m (a (m-3)+b (m-4) x)}{(m-4) (m-3) \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 40, normalized size = 0.6 \begin{align*}{\frac{ \left ( bmx+am-4\,bx-3\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( -3+m \right ) \left ( -4+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*(b*x+a)/(c*x^2)^(5/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.04902, size = 53, normalized size = 0.79 \begin{align*} \frac{b d^{m} x^{m}}{c^{\frac{5}{2}}{\left (m - 3\right )} x^{3}} + \frac{a d^{m} x^{m}}{c^{\frac{5}{2}}{\left (m - 4\right )} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.32064, size = 113, normalized size = 1.69 \begin{align*} \frac{\sqrt{c x^{2}}{\left (a m +{\left (b m - 4 \, b\right )} x - 3 \, a\right )} \left (d x\right )^{m}}{{\left (c^{3} m^{2} - 7 \, c^{3} m + 12 \, c^{3}\right )} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)*(a*m + (b*m - 4*b)*x - 3*a)*(d*x)^m/((c^3*m^2 - 7*c^3*m + 12*c^3)*x^5)

________________________________________________________________________________________

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*(b*x+a)/(c*x**2)**(5/2),x)

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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