3.1497 \(\int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0028276, 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} \[ -\frac{2 \sqrt{c+d x}}{\sqrt{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0078147, size = 30, normalized size = 1. \[ \frac{2 \sqrt{c+d x}}{\sqrt{a+b x} (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 27, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{dx+c}}{\sqrt{bx+a} \left ( ad-bc \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.2934, size = 92, normalized size = 3.07 \begin{align*} -\frac{2 \, \sqrt{b x + a} \sqrt{d x + c}}{a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.07062, size = 89, normalized size = 2.97 \begin{align*} -\frac{4 \, \sqrt{b d} b}{{\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 )}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*sqrt(b*d)*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)*abs(b))