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

Optimal result

Integrand size = 19, antiderivative size = 32 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx=-\frac {2 (c+d x)^{3/2}}{3 (b c-a d) (a+b x)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx=-\frac {2 (c+d x)^{3/2}}{3 (a+b x)^{3/2} (b c-a d)} \]

[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*} \text {integral}& = -\frac {2 (c+d x)^{3/2}}{3 (b c-a d) (a+b x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx=-\frac {2 (c+d x)^{3/2}}{3 (b c-a d) (a+b x)^{3/2}} \]

[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))

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

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	\(-\frac {\sqrt {d x +c}}{b \left (b x +a \right )^{\frac {3}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{2 b}\)	\(88\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx=-\frac {2 \, \sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{2}}}{3 \, {\left (a^{2} b c - a^{3} d + {\left (b^{3} c - a b^{2} d\right )} x^{2} + 2 \, {\left (a b^{2} c - a^{2} b d\right )} x\right )}} \]

[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)

Sympy [F]

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

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (26) = 52\).

Time = 0.36 (sec) , antiderivative size = 152, normalized size of antiderivative = 4.75 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx=-\frac {4 \, {\left (\sqrt {b d} b^{4} c^{2} d - 2 \, \sqrt {b d} a b^{3} c d^{2} + \sqrt {b d} a^{2} b^{2} d^{3} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} d\right )} {\left | b \right |}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} b^{2}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx=\frac {2\,{\left (c+d\,x\right )}^{3/2}}{\left (3\,a\,d-3\,b\,c\right )\,{\left (a+b\,x\right )}^{3/2}} \]

[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))