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

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

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Rubi [A]  time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.04, size = 53, normalized size = 0.68 \begin {gather*} -\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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.11, size = 63, normalized size = 0.81 \begin {gather*} \frac {2 \left (-8 d^2-2 d (d+e x)+3 (d+e x)^2\right ) \sqrt {2 c d (d+e x)-c (d+e x)^2}}{15 e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 56, normalized size = 0.72 \begin {gather*} \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 {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}\,{d x} \end {gather*}

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)

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maple [A]  time = 0.04, size = 44, normalized size = 0.56 \begin {gather*} -\frac {2 \left (-e x +d \right ) \left (3 e x +7 d \right ) \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{15 \sqrt {e x +d}\, e} \end {gather*}

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 = 1.56, size = 54, normalized size = 0.69 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.52, size = 69, normalized size = 0.88 \begin {gather*} \frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}}{5}-\frac {14\,d^2\,\sqrt {d+e\,x}}{15\,e^2}+\frac {8\,d\,x\,\sqrt {d+e\,x}}{15\,e}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c*d^2 - c*e^2*x^2)^(1/2)*((2*x^2*(d + e*x)^(1/2))/5 - (14*d^2*(d + e*x)^(1/2))/(15*e^2) + (8*d*x*(d + e*x)^(
1/2))/(15*e)))/(x + d/e)

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

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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