\(\int \frac {x}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx\) [751]

Optimal result

Integrand size = 20, antiderivative size = 80 \[ \int \frac {x}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {2 c \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {2 (b c-3 a d) \sqrt {a+b x}}{3 d (b c-a d)^2 \sqrt {c+d x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37} \[ \int \frac {x}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {2 \sqrt {a+b x} (b c-3 a d)}{3 d \sqrt {c+d x} (b c-a d)^2}-\frac {2 c \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]

[In]

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

[Out]

(-2*c*Sqrt[a + b*x])/(3*d*(b*c - a*d)*(c + d*x)^(3/2)) + (2*(b*c - 3*a*d)*Sqrt[a + b*x])/(3*d*(b*c - a*d)^2*Sq
rt[c + d*x])

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]

Rule 79

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {(b c-3 a d) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 d (b c-a d)} \\ & = -\frac {2 c \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {2 (b c-3 a d) \sqrt {a+b x}}{3 d (b c-a d)^2 \sqrt {c+d x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.58 \[ \int \frac {x}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {2 \sqrt {a+b x} (-2 a c+b c x-3 a d x)}{3 (b c-a d)^2 (c+d x)^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.68 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.52

method	result	size
default	\(-\frac {2 \sqrt {b x +a}\, \left (3 a d x -b c x +2 a c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} \left (a d -b c \right )^{2}}\)	\(42\)
gosper	\(-\frac {2 \sqrt {b x +a}\, \left (3 a d x -b c x +2 a c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(55\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.50 \[ \int \frac {x}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (2 \, a c - {\left (b c - 3 \, a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x/(d*x+c)^(5/2)/(b*x+a)^(1/2),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. 147 vs. \(2 (68) = 136\).

Time = 0.35 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.84 \[ \int \frac {x}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (b^{5} c d {\left | b \right |} - 3 \, a b^{4} d^{2} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}} - \frac {3 \, {\left (a b^{5} c d {\left | b \right |} - a^{2} b^{4} d^{2} {\left | b \right |}\right )}}{b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}} b} \]

[In]

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

[Out]

2/3*sqrt(b*x + a)*((b^5*c*d*abs(b) - 3*a*b^4*d^2*abs(b))*(b*x + a)/(b^4*c^2*d - 2*a*b^3*c*d^2 + a^2*b^2*d^3) -
3*(a*b^5*c*d*abs(b) - a^2*b^4*d^2*abs(b))/(b^4*c^2*d - 2*a*b^3*c*d^2 + a^2*b^2*d^3))/((b^2*c + (b*x + a)*b*d
- a*b*d)^(3/2)*b)

Mupad [B] (verification not implemented)

Time = 1.57 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.61 \[ \int \frac {x}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {x\,\left (6\,d\,a^2+2\,b\,c\,a\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}+\frac {4\,a^2\,c}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}-\frac {x^2\,\left (2\,b^2\,c-6\,a\,b\,d\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}\right )}{x^2\,\sqrt {a+b\,x}+\frac {c^2\,\sqrt {a+b\,x}}{d^2}+\frac {2\,c\,x\,\sqrt {a+b\,x}}{d}} \]

[In]

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

[Out]

-((c + d*x)^(1/2)*((x*(6*a^2*d + 2*a*b*c))/(3*d^2*(a*d - b*c)^2) + (4*a^2*c)/(3*d^2*(a*d - b*c)^2) - (x^2*(2*b
^2*c - 6*a*b*d))/(3*d^2*(a*d - b*c)^2)))/(x^2*(a + b*x)^(1/2) + (c^2*(a + b*x)^(1/2))/d^2 + (2*c*x*(a + b*x)^(
1/2))/d)