3.941 \(\int \frac{(c x^2)^{5/2} (a+b x)^n}{x^3} \, dx\)

Optimal. Leaf size=105 \[ \frac{a^2 c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^3 (n+1) x}-\frac{2 a c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^3 (n+2) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{n+3}}{b^3 (n+3) x} \]

[Out]

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

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Rubi [A]  time = 0.029299, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 43} \[ \frac{a^2 c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^3 (n+1) x}-\frac{2 a c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^3 (n+2) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{n+3}}{b^3 (n+3) x} \]

Antiderivative was successfully verified.

[In]

Int[((c*x^2)^(5/2)*(a + b*x)^n)/x^3,x]

[Out]

(a^2*c^2*Sqrt[c*x^2]*(a + b*x)^(1 + n))/(b^3*(1 + n)*x) - (2*a*c^2*Sqrt[c*x^2]*(a + b*x)^(2 + n))/(b^3*(2 + n)
*x) + (c^2*Sqrt[c*x^2]*(a + b*x)^(3 + n))/(b^3*(3 + 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 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{\left (c x^2\right )^{5/2} (a+b x)^n}{x^3} \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int x^2 (a+b x)^n \, dx}{x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \left (\frac{a^2 (a+b x)^n}{b^2}-\frac{2 a (a+b x)^{1+n}}{b^2}+\frac{(a+b x)^{2+n}}{b^2}\right ) \, dx}{x}\\ &=\frac{a^2 c^2 \sqrt{c x^2} (a+b x)^{1+n}}{b^3 (1+n) x}-\frac{2 a c^2 \sqrt{c x^2} (a+b x)^{2+n}}{b^3 (2+n) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{3+n}}{b^3 (3+n) x}\\ \end{align*}

Mathematica [A]  time = 0.0455615, size = 70, normalized size = 0.67 \[ \frac{c^3 x (a+b x)^{n+1} \left (2 a^2-2 a b (n+1) x+b^2 \left (n^2+3 n+2\right ) x^2\right )}{b^3 (n+1) (n+2) (n+3) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*x^2)^(5/2)*(a + b*x)^n)/x^3,x]

[Out]

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

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Maple [A]  time = 0.005, size = 83, normalized size = 0.8 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{2}{n}^{2}{x}^{2}+3\,{b}^{2}n{x}^{2}-2\,abnx+2\,{b}^{2}{x}^{2}-2\,abx+2\,{a}^{2} \right ) }{{x}^{5}{b}^{3} \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) } \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^3,x)

[Out]

(b*x+a)^(1+n)*(b^2*n^2*x^2+3*b^2*n*x^2-2*a*b*n*x+2*b^2*x^2-2*a*b*x+2*a^2)*(c*x^2)^(5/2)/x^5/b^3/(n^3+6*n^2+11*
n+6)

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Maxima [A]  time = 1.01093, size = 108, normalized size = 1.03 \begin{align*} \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} c^{\frac{5}{2}} x^{3} +{\left (n^{2} + n\right )} a b^{2} c^{\frac{5}{2}} x^{2} - 2 \, a^{2} b c^{\frac{5}{2}} n x + 2 \, a^{3} c^{\frac{5}{2}}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(5/2)*(b*x+a)^n/x^3,x, algorithm="maxima")

[Out]

((n^2 + 3*n + 2)*b^3*c^(5/2)*x^3 + (n^2 + n)*a*b^2*c^(5/2)*x^2 - 2*a^2*b*c^(5/2)*n*x + 2*a^3*c^(5/2))*(b*x + a
)^n/((n^3 + 6*n^2 + 11*n + 6)*b^3)

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Fricas [A]  time = 1.69188, size = 247, normalized size = 2.35 \begin{align*} -\frac{{\left (2 \, a^{2} b c^{2} n x - 2 \, a^{3} c^{2} -{\left (b^{3} c^{2} n^{2} + 3 \, b^{3} c^{2} n + 2 \, b^{3} c^{2}\right )} x^{3} -{\left (a b^{2} c^{2} n^{2} + a b^{2} c^{2} n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}\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^3,x, algorithm="fricas")

[Out]

-(2*a^2*b*c^2*n*x - 2*a^3*c^2 - (b^3*c^2*n^2 + 3*b^3*c^2*n + 2*b^3*c^2)*x^3 - (a*b^2*c^2*n^2 + a*b^2*c^2*n)*x^
2)*sqrt(c*x^2)*(b*x + a)^n/((b^3*n^3 + 6*b^3*n^2 + 11*b^3*n + 6*b^3)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x^{2}\right )^{\frac{5}{2}} \left (a + b x\right )^{n}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2)**(5/2)*(b*x+a)**n/x**3,x)

[Out]

Integral((c*x**2)**(5/2)*(a + b*x)**n/x**3, x)

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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^{3}}\,{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^3,x, algorithm="giac")

[Out]

integrate((c*x^2)^(5/2)*(b*x + a)^n/x^3, x)