3.15.66 \(\int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx\) [1466]

Optimal. Leaf size=32 \[ -\frac {2 (c+d x)^{3/2}}{3 (b c-a d) (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \begin {gather*} -\frac {2 (c+d x)^{3/2}}{3 (a+b x)^{3/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(a + b*x)^(5/2),x]

[Out]

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

Rule 37

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

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx &=-\frac {2 (c+d x)^{3/2}}{3 (b c-a d) (a+b x)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 32, normalized size = 1.00 \begin {gather*} -\frac {2 (c+d x)^{3/2}}{3 (b c-a d) (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs. \(2(26)=52\).
time = 0.16, size = 88, normalized size = 2.75

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 \left (b x +a \right )^{\frac {3}{2}} \left (a d -b c \right )}\)	\(27\)
default	\(-\frac {\sqrt {d x +c}}{b \left (b x +a \right )^{\frac {3}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{2 b}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/b*(d*x+c)^(1/2)/(b*x+a)^(3/2)+1/2*(a*d-b*c)/b*(-2/3*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(3/2)+4/3*d*(d*x+c)^(1
/2)/(-a*d+b*c)^2/(b*x+a)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (26) = 52\).
time = 0.74, size = 65, normalized size = 2.03 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{2}}}{3 \, {\left (a^{2} b c - a^{3} d + {\left (b^{3} c - a b^{2} d\right )} x^{2} + 2 \, {\left (a b^{2} c - a^{2} b d\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2)/(b*x+a)**(5/2),x)

[Out]

Integral(sqrt(c + d*x)/(a + b*x)**(5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (26) = 52\).
time = 1.09, size = 152, normalized size = 4.75 \begin {gather*} -\frac {4 \, {\left (\sqrt {b d} b^{4} c^{2} d - 2 \, \sqrt {b d} a b^{3} c d^{2} + \sqrt {b d} a^{2} b^{2} d^{3} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} d\right )} {\left | b \right |}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(5/2),x, algorithm="giac")

[Out]

-4/3*(sqrt(b*d)*b^4*c^2*d - 2*sqrt(b*d)*a*b^3*c*d^2 + sqrt(b*d)*a^2*b^2*d^3 + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*d)*abs(b)/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^2)^3*b^2)

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Mupad [B]
time = 0.72, size = 27, normalized size = 0.84 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{3/2}}{\left (3\,a\,d-3\,b\,c\right )\,{\left (a+b\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(1/2)/(a + b*x)^(5/2),x)

[Out]

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

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