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

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

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Rubi [A]  time = 0.00, antiderivative size = 30, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {2 \sqrt {c+d x}}{\sqrt {a+b x} (b c-a d)} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {c+d x}}{\sqrt {a+b x} (a d-b c)} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.03, size = 30, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {c+d x}}{\sqrt {a+b x} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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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fricas [A]  time = 0.92, size = 42, normalized size = 1.40 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 1.04, size = 66, normalized size = 2.20 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.01, size = 27, normalized size = 0.90 \begin {gather*} \frac {2 \sqrt {d x +c}}{\sqrt {b x +a}\, \left (a d -b c \right )} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> 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 details)Is a*d-b*c zero or nonzero?

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mupad [B]  time = 0.73, size = 26, normalized size = 0.87 \begin {gather*} \frac {2\,\sqrt {c+d\,x}}{\left (a\,d-b\,c\right )\,\sqrt {a+b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \end {gather*}

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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