3.9.14 \(\int \frac {x^2 \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx\) [814]

Optimal. Leaf size=28 \[ \frac {x^3 \sqrt {a+b x}}{3 \sqrt {-a-b x}} \]

[Out]

1/3*x^3*(b*x+a)^(1/2)/(-b*x-a)^(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 = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {23, 30} \begin {gather*} \frac {x^3 \sqrt {a+b x}}{3 \sqrt {-a-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*Sqrt[a + b*x])/Sqrt[-a - b*x],x]

[Out]

(x^3*Sqrt[a + b*x])/(3*Sqrt[-a - b*x])

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} \frac {x^3 \sqrt {a+b x}}{3 \sqrt {-a-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*Sqrt[a + b*x])/(3*Sqrt[-a - b*x])

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Maple [A]
time = 0.07, size = 23, normalized size = 0.82

method	result	size
gosper	\(\frac {x^{3} \sqrt {b x +a}}{3 \sqrt {-b x -a}}\)	\(23\)
default	\(-\frac {\sqrt {-b x -a}\, x^{3}}{3 \sqrt {b x +a}}\)	\(23\)
risch	\(-\frac {i \sqrt {\frac {-b x -a}{b x +a}}\, \sqrt {b x +a}\, x^{3}}{3 \sqrt {-b x -a}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [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(x^2*(b*x+a)^(1/2)/(-b*x-a)^(1/2),x, algorithm="fricas")

[Out]

0

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).
time = 1.95, size = 71, normalized size = 2.54 \begin {gather*} \frac {a^{2} \left (a + b x\right )^{\frac {3}{2}}}{b^{3} \sqrt {- a - b x}} - \frac {a \left (a + b x\right )^{\frac {5}{2}}}{b^{3} \sqrt {- a - b x}} + \frac {\left (a + b x\right )^{\frac {7}{2}}}{3 b^{3} \sqrt {- a - b x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(a + b*x)**(3/2)/(b**3*sqrt(-a - b*x)) - a*(a + b*x)**(5/2)/(b**3*sqrt(-a - b*x)) + (a + b*x)**(7/2)/(3*b
**3*sqrt(-a - b*x))

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Giac [C] Result contains complex when optimal does not.
time = 2.22, size = 33, normalized size = 1.18 \begin {gather*} -\frac {i \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2}\right )}}{3 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^2\,\sqrt {a+b\,x}}{\sqrt {-a-b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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