3.10.49 \(\int \frac {x (a+b x)^n}{\sqrt {c x^2}} \, dx\) [949]

Optimal. Leaf size=28 \[ \frac {x (a+b x)^{1+n}}{b (1+n) \sqrt {c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 32} \begin {gather*} \frac {x (a+b x)^{n+1}}{b (n+1) \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x)^n)/Sqrt[c*x^2],x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} \frac {x (a+b x)^{1+n}}{b (1+n) \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.13, size = 27, normalized size = 0.96

method	result	size
gosper	\(\frac {x \left (b x +a \right )^{1+n}}{b \left (1+n \right ) \sqrt {c \,x^{2}}}\)	\(27\)
risch	\(\frac {\left (b x +a \right ) x \left (b x +a \right )^{n}}{b \left (1+n \right ) \sqrt {c \,x^{2}}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)^n/(c*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 31, normalized size = 1.11 \begin {gather*} \frac {{\left (b \sqrt {c} x + a \sqrt {c}\right )} {\left (b x + a\right )}^{n}}{b c {\left (n + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.11, size = 33, normalized size = 1.18 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x + a\right )} {\left (b x + a\right )}^{n}}{{\left (b c n + b c\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n/(c*x^2)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n/(c*x^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [B]
time = 0.22, size = 36, normalized size = 1.29 \begin {gather*} \frac {\left (\frac {x^2}{n+1}+\frac {a\,x}{b\,\left (n+1\right )}\right )\,{\left (a+b\,x\right )}^n}{\sqrt {c\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a + b*x)^n)/(c*x^2)^(1/2),x)

[Out]

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

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