3.9.94 \(\int \frac {(c x^2)^{5/2} (a+b x)^n}{x^4} \, dx\)

Optimal. Leaf size=69 \[ \frac {c^2 \sqrt {c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac {a c^2 \sqrt {c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \begin {gather*} \frac {c^2 \sqrt {c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac {a c^2 \sqrt {c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c*x^2)^(5/2)*(a + b*x)^n)/x^4,x]

[Out]

-((a*c^2*Sqrt[c*x^2]*(a + b*x)^(1 + n))/(b^2*(1 + n)*x)) + (c^2*Sqrt[c*x^2]*(a + b*x)^(2 + n))/(b^2*(2 + 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^4} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x (a+b x)^n \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (-\frac {a (a+b x)^n}{b}+\frac {(a+b x)^{1+n}}{b}\right ) \, dx}{x}\\ &=-\frac {a c^2 \sqrt {c x^2} (a+b x)^{1+n}}{b^2 (1+n) x}+\frac {c^2 \sqrt {c x^2} (a+b x)^{2+n}}{b^2 (2+n) x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 46, normalized size = 0.67 \begin {gather*} \frac {c^3 x (a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2) \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((c*x^2)^(5/2)*(a + b*x)^n)/x^4,x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c x^2\right )^{5/2} (a+b x)^n}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((c*x^2)^(5/2)*(a + b*x)^n)/x^4,x]

[Out]

Defer[IntegrateAlgebraic][((c*x^2)^(5/2)*(a + b*x)^n)/x^4, x]

________________________________________________________________________________________

fricas [A]  time = 1.05, size = 76, normalized size = 1.10 \begin {gather*} \frac {{\left (a b c^{2} n x - a^{2} c^{2} + {\left (b^{2} c^{2} n + b^{2} c^{2}\right )} x^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(5/2)*(b*x+a)^n/x^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c x^{2}\right )^{\frac {5}{2}} {\left (b x + a\right )}^{n}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(5/2)*(b*x+a)^n/x^4,x, algorithm="giac")

[Out]

integrate((c*x^2)^(5/2)*(b*x + a)^n/x^4, x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 46, normalized size = 0.67 \begin {gather*} -\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (-x n b -b x +a \right ) \left (b x +a \right )^{n +1}}{\left (n^{2}+3 n +2\right ) b^{2} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(5/2)*(b*x+a)^n/x^4,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.45, size = 51, normalized size = 0.74 \begin {gather*} \frac {{\left (b^{2} c^{\frac {5}{2}} {\left (n + 1\right )} x^{2} + a b c^{\frac {5}{2}} n x - a^{2} c^{\frac {5}{2}}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(5/2)*(b*x+a)^n/x^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.24, size = 94, normalized size = 1.36 \begin {gather*} \frac {{\left (a+b\,x\right )}^n\,\left (\frac {c^2\,x^2\,\sqrt {c\,x^2}\,\left (n+1\right )}{n^2+3\,n+2}-\frac {a^2\,c^2\,\sqrt {c\,x^2}}{b^2\,\left (n^2+3\,n+2\right )}+\frac {a\,c^2\,n\,x\,\sqrt {c\,x^2}}{b\,\left (n^2+3\,n+2\right )}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*x^2)^(5/2)*(a + b*x)^n)/x^4,x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {a^{n} c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{2 x^{3}} & \text {for}\: b = 0 \\\int \frac {\left (c x^{2}\right )^{\frac {5}{2}}}{x^{4} \left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {\left (c x^{2}\right )^{\frac {5}{2}}}{x^{4} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\- \frac {a^{2} c^{\frac {5}{2}} \left (a + b x\right )^{n} \left (x^{2}\right )^{\frac {5}{2}}}{b^{2} n^{2} x^{5} + 3 b^{2} n x^{5} + 2 b^{2} x^{5}} + \frac {a b c^{\frac {5}{2}} n x \left (a + b x\right )^{n} \left (x^{2}\right )^{\frac {5}{2}}}{b^{2} n^{2} x^{5} + 3 b^{2} n x^{5} + 2 b^{2} x^{5}} + \frac {b^{2} c^{\frac {5}{2}} n x^{2} \left (a + b x\right )^{n} \left (x^{2}\right )^{\frac {5}{2}}}{b^{2} n^{2} x^{5} + 3 b^{2} n x^{5} + 2 b^{2} x^{5}} + \frac {b^{2} c^{\frac {5}{2}} x^{2} \left (a + b x\right )^{n} \left (x^{2}\right )^{\frac {5}{2}}}{b^{2} n^{2} x^{5} + 3 b^{2} n x^{5} + 2 b^{2} x^{5}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2)**(5/2)*(b*x+a)**n/x**4,x)

[Out]

Piecewise((a**n*c**(5/2)*(x**2)**(5/2)/(2*x**3), Eq(b, 0)), (Integral((c*x**2)**(5/2)/(x**4*(a + b*x)**2), x),
Eq(n, -2)), (Integral((c*x**2)**(5/2)/(x**4*(a + b*x)), x), Eq(n, -1)), (-a**2*c**(5/2)*(a + b*x)**n*(x**2)**
(5/2)/(b**2*n**2*x**5 + 3*b**2*n*x**5 + 2*b**2*x**5) + a*b*c**(5/2)*n*x*(a + b*x)**n*(x**2)**(5/2)/(b**2*n**2*
x**5 + 3*b**2*n*x**5 + 2*b**2*x**5) + b**2*c**(5/2)*n*x**2*(a + b*x)**n*(x**2)**(5/2)/(b**2*n**2*x**5 + 3*b**2
*n*x**5 + 2*b**2*x**5) + b**2*c**(5/2)*x**2*(a + b*x)**n*(x**2)**(5/2)/(b**2*n**2*x**5 + 3*b**2*n*x**5 + 2*b**
2*x**5), True))

________________________________________________________________________________________