\(\int \frac {\sqrt {a+b x}}{(c+d x)^{5/2}} \, dx\) [594]

Optimal result

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

[Out]

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

[In]

Int[Sqrt[a + b*x]/(c + d*x)^(5/2),x]

[Out]

(2*(a + b*x)^(3/2))/(3*(b*c - a*d)*(c + d*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 (a+b x)^{3/2}}{3 (b c-a d) (c+d x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {2 (a+b x)^{3/2}}{3 (b c-a d) (c+d x)^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

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

[In]

int((b*x+a)^(1/2)/(d*x+c)^(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 = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c}}{3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((b*x+a)^(1/2)/(d*x+c)^(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 [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{4} d}{3 \, {\left (b c d {\left | b \right |} - a d^{2} {\left | b \right |}\right )} {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.06 \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=-\frac {\left (\frac {2\,a\,\sqrt {a+b\,x}}{3\,a\,d^3-3\,b\,c\,d^2}+\frac {2\,b\,x\,\sqrt {a+b\,x}}{3\,a\,d^3-3\,b\,c\,d^2}\right )\,\sqrt {c+d\,x}}{x^2-\frac {3\,b\,c^3-3\,a\,c^2\,d}{3\,a\,d^3-3\,b\,c\,d^2}+\frac {6\,c\,d\,x\,\left (a\,d-b\,c\right )}{3\,a\,d^3-3\,b\,c\,d^2}} \]

[In]

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

[Out]

-(((2*a*(a + b*x)^(1/2))/(3*a*d^3 - 3*b*c*d^2) + (2*b*x*(a + b*x)^(1/2))/(3*a*d^3 - 3*b*c*d^2))*(c + d*x)^(1/2
))/(x^2 - (3*b*c^3 - 3*a*c^2*d)/(3*a*d^3 - 3*b*c*d^2) + (6*c*d*x*(a*d - b*c))/(3*a*d^3 - 3*b*c*d^2))