3.744 \(\int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=30 \[ \frac{2 \sqrt{a+b x}}{\sqrt{c+d x} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0030084, antiderivative size = 30, normalized size of antiderivative = 1., 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} \[ \frac{2 \sqrt{a+b x}}{\sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x]*(c + d*x)^(3/2)),x]

[Out]

(2*Sqrt[a + b*x])/((b*c - a*d)*Sqrt[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]

Rubi steps

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

Mathematica [A]  time = 0.0069611, size = 30, normalized size = 1. \[ \frac{2 \sqrt{a+b x}}{\sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x]*(c + d*x)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0., size = 27, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{bx+a}}{\sqrt{dx+c} \left ( ad-bc \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/2)/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.9238, size = 90, normalized size = 3. \begin{align*} \frac{2 \, \sqrt{b x + a} \sqrt{d x + c}}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/2)/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(1/2)/(d*x+c)**(3/2),x)

[Out]

Integral(1/(sqrt(a + b*x)*(c + d*x)**(3/2)), x)

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Giac [A]  time = 1.47239, size = 63, normalized size = 2.1 \begin{align*} \frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (b c{\left | b \right |} - a d{\left | b \right |}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/2)/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

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