3.864 \(\int \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2} \, dx\)

Optimal. Leaf size=78 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{5 c e \sqrt{d+e x}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{3/2}}{15 c e (d+e x)^{3/2}} \]

[Out]

(-8*d*(c*d^2 - c*e^2*x^2)^(3/2))/(15*c*e*(d + e*x)^(3/2)) - (2*(c*d^2 - c*e^2*x^2)^(3/2))/(5*c*e*Sqrt[d + e*x]
)

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Rubi [A]  time = 0.0279207, 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} \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{5 c e \sqrt{d+e x}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{3/2}}{15 c e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

(-8*d*(c*d^2 - c*e^2*x^2)^(3/2))/(15*c*e*(d + e*x)^(3/2)) - (2*(c*d^2 - c*e^2*x^2)^(3/2))/(5*c*e*Sqrt[d + e*x]
)

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 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2} \, dx &=-\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{5 c e \sqrt{d+e x}}+\frac{1}{5} (4 d) \int \frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{d+e x}} \, dx\\ &=-\frac{8 d \left (c d^2-c e^2 x^2\right )^{3/2}}{15 c e (d+e x)^{3/2}}-\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{5 c e \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0432887, size = 53, normalized size = 0.68 \[ -\frac{2 \left (7 d^2-4 d e x-3 e^2 x^2\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{15 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

(-2*(7*d^2 - 4*d*e*x - 3*e^2*x^2)*Sqrt[c*(d^2 - e^2*x^2)])/(15*e*Sqrt[d + e*x])

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Maple [A]  time = 0.045, size = 44, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 3\,ex+7\,d \right ) }{15\,e}\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(-e*x+d)*(3*e*x+7*d)*(-c*e^2*x^2+c*d^2)^(1/2)/e/(e*x+d)^(1/2)

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Maxima [A]  time = 1.05922, size = 73, normalized size = 0.94 \begin{align*} \frac{2 \,{\left (3 \, \sqrt{c} e^{2} x^{2} + 4 \, \sqrt{c} d e x - 7 \, \sqrt{c} d^{2}\right )}{\left (e x + d\right )} \sqrt{-e x + d}}{15 \,{\left (e^{2} x + d e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

2/15*(3*sqrt(c)*e^2*x^2 + 4*sqrt(c)*d*e*x - 7*sqrt(c)*d^2)*(e*x + d)*sqrt(-e*x + d)/(e^2*x + d*e)

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Fricas [A]  time = 2.07991, size = 120, normalized size = 1.54 \begin{align*} \frac{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (3 \, e^{2} x^{2} + 4 \, d e x - 7 \, d^{2}\right )} \sqrt{e x + d}}{15 \,{\left (e^{2} x + d e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

2/15*sqrt(-c*e^2*x^2 + c*d^2)*(3*e^2*x^2 + 4*d*e*x - 7*d^2)*sqrt(e*x + d)/(e^2*x + d*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="giac")

[Out]

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