3.4.68 \(\int \frac {x^{-1+m} (2 a m+b (-1+2 m) x)}{2 (a+b x)^{3/2}} \, dx\) [368]

Optimal. Leaf size=13 \[ \frac {x^m}{\sqrt {a+b x}} \]

[Out]

x^m/(b*x+a)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {12, 75} \begin {gather*} \frac {x^m}{\sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^(-1 + m)*(2*a*m + b*(-1 + 2*m)*x))/(2*(a + b*x)^(3/2)),x]

[Out]

x^m/Sqrt[a + b*x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

\begin {align*} \int \frac {x^{-1+m} (2 a m+b (-1+2 m) x)}{2 (a+b x)^{3/2}} \, dx &=\frac {1}{2} \int \frac {x^{-1+m} (2 a m+b (-1+2 m) x)}{(a+b x)^{3/2}} \, dx\\ &=\frac {x^m}{\sqrt {a+b x}}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 13, normalized size = 1.00 \begin {gather*} \frac {x^m}{\sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(-1 + m)*(2*a*m + b*(-1 + 2*m)*x))/(2*(a + b*x)^(3/2)),x]

[Out]

x^m/Sqrt[a + b*x]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 35.47, size = 69, normalized size = 5.31 \begin {gather*} \frac {a \left (1+m\right ) x^m \text {hyper}\left [\left \{\frac {3}{2},m\right \},\left \{1+m\right \},\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a}\right ]+\frac {b \left (-1+2 m\right ) x^{1+m} \text {hyper}\left [\left \{\frac {3}{2},1+m\right \},\left \{2+m\right \},\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a}\right ]}{2}}{a^{\frac {3}{2}} \left (1+m\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(x^(-1 + m)*(2*a*m + b*(-1 + 2*m)*x))/(2*(a + b*x)^(3/2)),x]')

[Out]

(a (1 + m) x ^ m hyper[{3 / 2, m}, {1 + m}, b x exp_polar[I Pi] / a] + b (-1 + 2 m) x ^ (1 + m) hyper[{3 / 2,
1 + m}, {2 + m}, b x exp_polar[I Pi] / a] / 2) / (a ^ (3 / 2) (1 + m))

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Maple [A]
time = 0.09, size = 12, normalized size = 0.92

method	result	size
gosper	\(\frac {x^{m}}{\sqrt {b x +a}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*x^(-1+m)*(2*a*m+b*(-1+2*m)*x)/(b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

x^m/(b*x+a)^(1/2)

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Maxima [A]
time = 0.31, size = 11, normalized size = 0.85 \begin {gather*} \frac {x^{m}}{\sqrt {b x + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*x^(-1+m)*(2*a*m+b*(-1+2*m)*x)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

x^m/sqrt(b*x + a)

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Fricas [A]
time = 0.31, size = 14, normalized size = 1.08 \begin {gather*} \frac {x x^{m - 1}}{\sqrt {b x + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*x^(-1+m)*(2*a*m+b*(-1+2*m)*x)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

x*x^(m - 1)/sqrt(b*x + a)

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Sympy [C] Result contains complex when optimal does not.
time = 39.31, size = 78, normalized size = 6.00 \begin {gather*} \frac {m x^{m} \Gamma \left (m\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, m \\ m + 1 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (m + 1\right )} + \frac {b x x^{m} \left (2 m - 1\right ) \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (m + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*x**(-1+m)*(2*a*m+b*(-1+2*m)*x)/(b*x+a)**(3/2),x)

[Out]

m*x**m*gamma(m)*hyper((3/2, m), (m + 1,), b*x*exp_polar(I*pi)/a)/(sqrt(a)*gamma(m + 1)) + b*x*x**m*(2*m - 1)*g
amma(m + 1)*hyper((3/2, m + 1), (m + 2,), b*x*exp_polar(I*pi)/a)/(2*a**(3/2)*gamma(m + 2))

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Giac [F] N/A
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(1/2*x^(-1+m)*(2*a*m+b*(-1+2*m)*x)/(b*x+a)^(3/2),x)

[Out]

Could not integrate

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Mupad [B]
time = 0.41, size = 11, normalized size = 0.85 \begin {gather*} \frac {x^m}{\sqrt {a+b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(m - 1)*(2*a*m + b*x*(2*m - 1)))/(2*(a + b*x)^(3/2)),x)

[Out]

x^m/(a + b*x)^(1/2)

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