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

Optimal. Leaf size=21 \[ -\frac {2}{b d \sqrt {d (a+b x)+c}} \]

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Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32} \begin {gather*} -\frac {2}{b d \sqrt {d (a+b x)+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*(a + b*x))^(-3/2),x]

[Out]

-2/(b*d*Sqrt[c + d*(a + b*x)])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 33

Int[((a_.) + (b_.)*(u_))^(m_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x]
/; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {2}{b d \sqrt {d (a+b x)+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*(a + b*x))^(-3/2),x]

[Out]

-2/(b*d*Sqrt[c + d*(a + b*x)])

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IntegrateAlgebraic [A]  time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {2}{b d \sqrt {a d+b d x+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(b*d*Sqrt[c + a*d + b*d*x])

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fricas [A]  time = 1.57, size = 34, normalized size = 1.62 \begin {gather*} -\frac {2 \, \sqrt {b d x + a d + c}}{b^{2} d^{2} x + a b d^{2} + b c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.65, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2}{\sqrt {b d x + a d + c} b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(b*d*x + a*d + c)*b*d)

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maple [A]  time = 0.00, size = 20, normalized size = 0.95 \begin {gather*} -\frac {2}{\sqrt {b d x +a d +c}\, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.00, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2}{\sqrt {{\left (b x + a\right )} d + c} b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt((b*x + a)*d + c)*b*d)

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mupad [B]  time = 0.13, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2}{b\,d\,\sqrt {c+d\,\left (a+b\,x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c + d*(a + b*x))^(3/2),x)

[Out]

-2/(b*d*(c + d*(a + b*x))^(1/2))

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sympy [A]  time = 1.79, size = 58, normalized size = 2.76 \begin {gather*} \begin {cases} \frac {x}{c^{\frac {3}{2}}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x}{\left (a d + c\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \\\frac {x}{c^{\frac {3}{2}}} & \text {for}\: d = 0 \\- \frac {2 \sqrt {a d + b d x + c}}{a b d^{2} + b^{2} d^{2} x + b c d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/c**(3/2), Eq(b, 0) & Eq(d, 0)), (x/(a*d + c)**(3/2), Eq(b, 0)), (x/c**(3/2), Eq(d, 0)), (-2*sqrt(
a*d + b*d*x + c)/(a*b*d**2 + b**2*d**2*x + b*c*d), True))

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