3.10.43 \(\int \frac {(c x^2)^{5/2} (a+b x)^n}{x^5} \, dx\) [943]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \begin {gather*} \frac {c^2 \sqrt {c x^2} (a+b x)^{n+1}}{b (n+1) x} \end {gather*}

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*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.94 \begin {gather*} \frac {c^3 x (a+b x)^{1+n}}{b (1+n) \sqrt {c x^2}} \end {gather*}

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 = 0.12, size = 29, normalized size = 0.88

method	result	size
gosper	\(\frac {\left (b x +a \right )^{1+n} \left (c \,x^{2}\right )^{\frac {5}{2}}}{b \left (1+n \right ) x^{5}}\)	\(29\)
risch	\(\frac {c^{2} \sqrt {c \,x^{2}}\, \left (b x +a \right ) \left (b x +a \right )^{n}}{x b \left (1+n \right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 28, normalized size = 0.85 \begin {gather*} \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 {gather*}

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 = 0.76, size = 37, normalized size = 1.12 \begin {gather*} \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 {gather*}

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.23, size = 49, normalized size = 1.48 \begin {gather*} \frac {\left (\frac {c^2\,x\,\sqrt {c\,x^2}}{n+1}+\frac {a\,c^2\,\sqrt {c\,x^2}}{b\,\left (n+1\right )}\right )\,{\left (a+b\,x\right )}^n}{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^5,x)

[Out]

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

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