3.2030 \(\int \sqrt {d+e x} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2} \, dx\)

Optimal. Leaf size=109 \[ \frac {4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt {d+e x}} \]

[Out]

4/15*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2/(e*x+d)^(3/2)+2/5*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(3/2)/c/d/(e*x+d)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac {4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

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

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rule 656

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 55, normalized size = 0.50 \[ \frac {2 ((d+e x) (a e+c d x))^{3/2} \left (c d (5 d+3 e x)-2 a e^2\right )}{15 c^2 d^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-2*a*e^2 + c*d*(5*d + 3*e*x)))/(15*c^2*d^2*(d + e*x)^(3/2))

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fricas [A]  time = 0.92, size = 102, normalized size = 0.94 \[ \frac {2 \, {\left (3 \, c^{2} d^{2} e x^{2} + 5 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (5 \, c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{15 \, {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*c^2*d^2*e*x^2 + 5*a*c*d^2*e - 2*a^2*e^3 + (5*c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x)*sqrt(e*x + d)/(c^2*d^2*e*x + c^2*d^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 69, normalized size = 0.63 \[ -\frac {2 \left (c d x +a e \right ) \left (-3 c d e x +2 a \,e^{2}-5 c \,d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{15 \sqrt {e x +d}\, c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(c*d*x+a*e)*(-3*c*d*e*x+2*a*e^2-5*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/c^2/d^2/(e*x+d)^(1/2)

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maxima [A]  time = 1.33, size = 83, normalized size = 0.76 \[ \frac {2 \, {\left (3 \, c^{2} d^{2} e x^{2} + 5 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (5 \, c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{15 \, {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*c^2*d^2*e*x^2 + 5*a*c*d^2*e - 2*a^2*e^3 + (5*c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^2*
d^2*e*x + c^2*d^3)

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mupad [B]  time = 0.80, size = 121, normalized size = 1.11 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}}{5}-\frac {\left (4\,a^2\,e^3-10\,a\,c\,d^2\,e\right )\,\sqrt {d+e\,x}}{15\,c^2\,d^2\,e}+\frac {x\,\left (10\,c^2\,d^3+2\,a\,c\,d\,e^2\right )\,\sqrt {d+e\,x}}{15\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*x^2*(d + e*x)^(1/2))/5 - ((4*a^2*e^3 - 10*a*c*d^2*e)*(d + e
*x)^(1/2))/(15*c^2*d^2*e) + (x*(10*c^2*d^3 + 2*a*c*d*e^2)*(d + e*x)^(1/2))/(15*c^2*d^2*e)))/(x + d/e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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