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

Optimal. Leaf size=32 \[ \frac{2 (a+b x)^{3/2}}{3 (c+d x)^{3/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0032584, antiderivative size = 32, 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} \[ \frac{2 (a+b x)^{3/2}}{3 (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[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*} \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}}\\ \end{align*}

Mathematica [A]  time = 0.0097221, size = 32, normalized size = 1. \[ \frac{2 (a+b x)^{3/2}}{3 (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[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))

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Maple [A]  time = 0.003, size = 27, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,ad-3\,bc} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(b*x+a)^(3/2)/(d*x+c)^(3/2)/(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((b*x+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.87929, size = 139, normalized size = 4.34 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x}}{\left (c + d x\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.31904, size = 90, normalized size = 2.81 \begin{align*} -\frac{{\left (b x + a\right )}^{\frac{3}{2}} b^{4} d}{24 \,{\left (b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}\right )}{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(b*x + a)^(3/2)*b^4*d/((b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6)*(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2)
)