3.943 \(\int \frac{(c x^2)^{5/2} (a+b x)^n}{x^5} \, dx\)

Optimal. Leaf size=33 \[ \frac{c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]

[Out]

(c^2*Sqrt[c*x^2]*(a + b*x)^(1 + n))/(b*(1 + n)*x)

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Rubi [A]  time = 0.0064947, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ \frac{c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*Sqrt[c*x^2]*(a + b*x)^(1 + n))/(b*(1 + n)*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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.016853, size = 31, normalized size = 0.94 \[ \frac{c^3 x (a+b x)^{n+1}}{b (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*x*(a + b*x)^(1 + n))/(b*(1 + n)*Sqrt[c*x^2])

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Maple [A]  time = 0.002, size = 29, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n}}{b \left ( 1+n \right ){x}^{5}} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02733, size = 38, normalized size = 1.15 \begin{align*} \frac{{\left (b c^{\frac{5}{2}} x + a c^{\frac{5}{2}}\right )}{\left (b x + a\right )}^{n}}{b{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57848, size = 77, normalized size = 2.33 \begin{align*} \frac{{\left (b c^{2} x + a c^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b n + b\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*c^2*x + a*c^2)*sqrt(c*x^2)*(b*x + a)^n/((b*n + b)*x)

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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