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

Optimal. Leaf size=78 \[ -\frac{8 d \sqrt{c d^2-c e^2 x^2}}{3 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{3 c e} \]

[Out]

(-8*d*Sqrt[c*d^2 - c*e^2*x^2])/(3*c*e*Sqrt[d + e*x]) - (2*Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2])/(3*c*e)

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Rubi [A]  time = 0.0301264, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {657, 649} \[ -\frac{8 d \sqrt{c d^2-c e^2 x^2}}{3 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{3 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*d*Sqrt[c*d^2 - c*e^2*x^2])/(3*c*e*Sqrt[d + e*x]) - (2*Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2])/(3*c*e)

Rule 657

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*(m + 2*p + 1)), x] + Dist[(2*c*d*Simplify[m + p])/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^
2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p]
, 0]

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}}{\sqrt{c d^2-c e^2 x^2}} \, dx &=-\frac{2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{3 c e}+\frac{1}{3} (4 d) \int \frac{\sqrt{d+e x}}{\sqrt{c d^2-c e^2 x^2}} \, dx\\ &=-\frac{8 d \sqrt{c d^2-c e^2 x^2}}{3 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{3 c e}\\ \end{align*}

Mathematica [A]  time = 0.0481572, size = 47, normalized size = 0.6 \[ -\frac{2 (d-e x) \sqrt{d+e x} (5 d+e x)}{3 e \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d - e*x)*Sqrt[d + e*x]*(5*d + e*x))/(3*e*Sqrt[c*(d^2 - e^2*x^2)])

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Maple [A]  time = 0.042, size = 43, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( ex+5\,d \right ) }{3\,e}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}}}} \end{align*}

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)^(1/2),x)

[Out]

-2/3*(-e*x+d)*(e*x+5*d)*(e*x+d)^(1/2)/e/(-c*e^2*x^2+c*d^2)^(1/2)

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Maxima [A]  time = 1.04609, size = 46, normalized size = 0.59 \begin{align*} \frac{2 \,{\left (e^{2} x^{2} + 4 \, d e x - 5 \, d^{2}\right )}}{3 \, \sqrt{-e x + d} \sqrt{c} e} \end{align*}

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)^(1/2),x, algorithm="maxima")

[Out]

2/3*(e^2*x^2 + 4*d*e*x - 5*d^2)/(sqrt(-e*x + d)*sqrt(c)*e)

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Fricas [A]  time = 2.13827, size = 101, normalized size = 1.29 \begin{align*} -\frac{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (e x + 5 \, d\right )} \sqrt{e x + d}}{3 \,{\left (c e^{2} x + c d e\right )}} \end{align*}

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)^(1/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(-c*e^2*x^2 + c*d^2)*(e*x + 5*d)*sqrt(e*x + d)/(c*e^2*x + c*d*e)

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

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)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{-c e^{2} x^{2} + c d^{2}}}\,{d x} \end{align*}

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)^(1/2),x, algorithm="giac")

[Out]

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