3.7.59 \(\int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [659]

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

[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 = 0.06, 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 = {808, 662} \begin {gather*} \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}} \end {gather*}

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 662

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 808

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 = 0.04, size = 53, normalized size = 0.42 \begin {gather*} \frac {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}} \end {gather*}

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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Maple [A]
time = 0.14, size = 49, normalized size = 0.39

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-c d g x +2 a e g -3 c d f \right )}{3 \sqrt {e x +d}\, c^{2} d^{2}}\)	\(49\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-c d g x +2 a e g -3 c d f \right ) \sqrt {e x +d}}{3 c^{2} d^{2} \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 67, normalized size = 0.54 \begin {gather*} \frac {2 \, \sqrt {c d x + a e} f}{c d} + \frac {2 \, {\left (c^{2} d^{2} x^{2} - a c d x e - 2 \, a^{2} e^{2}\right )} g}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} \end {gather*}

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

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Fricas [A]
time = 4.72, size = 74, normalized size = 0.59 \begin {gather*} \frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d g x + 3 \, c d f - 2 \, a g e\right )} \sqrt {x e + d}}{3 \, {\left (c^{2} d^{2} x e + c^{2} d^{3}\right )}} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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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Giac [A]
time = 1.43, size = 160, normalized size = 1.28 \begin {gather*} \frac {2 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} g e^{\left (-3\right )}}{3 \, c^{2} d^{2}} + \frac {2 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} {\left (c d f - a g e\right )} e^{\left (-1\right )}}{c^{2} d^{2}} + \frac {2 \, {\left (\sqrt {-c d^{2} e + a e^{3}} c d^{2} g - 3 \, \sqrt {-c d^{2} e + a e^{3}} c d f e + 2 \, \sqrt {-c d^{2} e + a e^{3}} a g e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2} d^{2}} \end {gather*}

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]

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

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Mupad [B]
time = 3.23, size = 88, normalized size = 0.70 \begin {gather*} -\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}} \end {gather*}

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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