3.8.30 \(\int \frac {(d+e x)^{3/2}}{(c d^2-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=36 \[ \frac {2 \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {649} \begin {gather*} \frac {2 \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(2*Sqrt[d + e*x])/(c*e*Sqrt[c*d^2 - c*e^2*x^2])

Rule 649

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 35, normalized size = 0.97 \begin {gather*} \frac {2 \sqrt {d+e x}}{c e \sqrt {c \left (d^2-e^2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(2*Sqrt[d + e*x])/(c*e*Sqrt[c*(d^2 - e^2*x^2)])

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IntegrateAlgebraic [A]  time = 0.74, size = 50, normalized size = 1.39 \begin {gather*} -\frac {2 \sqrt {2 c d (d+e x)-c (d+e x)^2}}{c^2 e (e x-d) \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2*c*d*(d + e*x) - c*(d + e*x)^2])/(c^2*e*(-d + e*x)*Sqrt[d + e*x])

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fricas [A]  time = 0.39, size = 48, normalized size = 1.33 \begin {gather*} -\frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{c^{2} e^{3} x^{2} - c^{2} d^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-2*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(c^2*e^3*x^2 - c^2*d^2*e)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.05, size = 36, normalized size = 1.00 \begin {gather*} \frac {2 \left (-e x +d \right ) \left (e x +d \right )^{\frac {3}{2}}}{\left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

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

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maxima [A]  time = 1.48, size = 16, normalized size = 0.44 \begin {gather*} \frac {2}{\sqrt {-e x + d} c^{\frac {3}{2}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

2/(sqrt(-e*x + d)*c^(3/2)*e)

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mupad [B]  time = 0.65, size = 50, normalized size = 1.39 \begin {gather*} \frac {2\,\sqrt {c\,d^2-c\,e^2\,x^2}\,\sqrt {d+e\,x}}{e\,\left (c^2\,d^2-c^2\,e^2\,x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(c*d^2 - c*e^2*x^2)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Integral((d + e*x)**(3/2)/(-c*(-d + e*x)*(d + e*x))**(3/2), x)

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