3.659 \(\int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=125 \[ \frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^2 d^2 e \sqrt {d+e x}} \]

[Out]

-2/3*(2*a*e^2*g-c*d*(-d*g+3*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/e/(e*x+d)^(1/2)+2/3*g*(e*x+d
)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e

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Rubi [A]  time = 0.09, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {794, 648} \[ \frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^2 d^2 e \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 794

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

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 g \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d e}+\frac {1}{3} \left (3 f-\frac {d g}{e}-\frac {2 a e g}{c d}\right ) \int \frac {\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 (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e \sqrt {d+e x}}+\frac {2 g \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d e}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 53, normalized size = 0.42 \[ \frac {2 \sqrt {(d+e x) (a e+c d x)} (c d (3 f+g x)-2 a e g)}{3 c^2 d^2 \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.05, size = 71, normalized size = 0.57 \[ \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x + 3 \, c d f - 2 \, a e g\right )} \sqrt {e x + d}}{3 \, {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(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]

2/3*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*g*x + 3*c*d*f - 2*a*e*g)*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 \[ \int \frac {\sqrt {e x + d} {\left (g x + f\right )}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 0.01, size = 67, normalized size = 0.54 \[ -\frac {2 \left (c d x +a e \right ) \left (-c d g x +2 a e g -3 c d f \right ) \sqrt {e x +d}}{3 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(c*d*x+a*e)*(-c*d*g*x+2*a*e*g-3*c*d*f)*(e*x+d)^(1/2)/c^2/d^2/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

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maxima [A]  time = 0.55, size = 65, normalized size = 0.52 \[ \frac {2 \, \sqrt {c d x + a e} f}{c d} + \frac {2 \, {\left (c^{2} d^{2} x^{2} - a c d e x - 2 \, a^{2} e^{2}\right )} g}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(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*sqrt(c*d*x + a*e)*f/(c*d) + 2/3*(c^2*d^2*x^2 - a*c*d*e*x - 2*a^2*e^2)*g/(sqrt(c*d*x + a*e)*c^2*d^2)

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mupad [B]  time = 3.23, size = 88, normalized size = 0.70 \[ -\frac {\left (\frac {\left (4\,a\,e\,g-6\,c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e}-\frac {2\,g\,x\,\sqrt {d+e\,x}}{3\,c\,d\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x+\frac {d}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((((4*a*e*g - 6*c*d*f)*(d + e*x)^(1/2))/(3*c^2*d^2*e) - (2*g*x*(d + e*x)^(1/2))/(3*c*d*e))*(x*(a*e^2 + c*d^2)
+ a*d*e + c*d*e*x^2)^(1/2))/(x + d/e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(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)*(f + g*x)/sqrt((d + e*x)*(a*e + c*d*x)), x)

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