3.23.34 \(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx\) [2234]

Optimal. Leaf size=118 \[ -\frac {2 (5 c e f-c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt {d+e x}} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {808, 662} \begin {gather*} -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g-c d g+5 c e f)}{15 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.06, size = 76, normalized size = 0.64 \begin {gather*} \frac {2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))} (-2 b e g+c (5 e f+2 d g+3 e g x))}{15 c^2 e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, size = 71, normalized size = 0.60

method	result	size
default	\(-\frac {2 \left (c e x +b e -c d \right ) \left (-3 c e g x +2 b e g -2 c d g -5 c e f \right ) \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}}{15 c^{2} e^{2} \sqrt {e x +d}}\)	\(71\)
gosper	\(-\frac {2 \left (c e x +b e -c d \right ) \left (-3 c e g x +2 b e g -2 c d g -5 c e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{15 c^{2} e^{2} \sqrt {e x +d}}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 115, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (c x e - c d + b e\right )} \sqrt {-c x e + c d - b e} f e^{\left (-1\right )}}{3 \, c} + \frac {2 \, {\left (3 \, c^{2} x^{2} e^{2} - 2 \, c^{2} d^{2} + 4 \, b c d e - 2 \, b^{2} e^{2} - {\left (c^{2} d e - b c e^{2}\right )} x\right )} \sqrt {-c x e + c d - b e} g e^{\left (-2\right )}}{15 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(c*x*e - c*d + b*e)*sqrt(-c*x*e + c*d - b*e)*f*e^(-1)/c + 2/15*(3*c^2*x^2*e^2 - 2*c^2*d^2 + 4*b*c*d*e - 2*
b^2*e^2 - (c^2*d*e - b*c*e^2)*x)*sqrt(-c*x*e + c*d - b*e)*g*e^(-2)/c^2

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Fricas [A]
time = 2.71, size = 126, normalized size = 1.07 \begin {gather*} -\frac {2 \, {\left (2 \, c^{2} d^{2} g - {\left (3 \, c^{2} g x^{2} + 5 \, b c f - 2 \, b^{2} g + {\left (5 \, c^{2} f + b c g\right )} x\right )} e^{2} + {\left (c^{2} d g x + 5 \, c^{2} d f - 4 \, b c d g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}}{15 \, {\left (c^{2} x e^{3} + c^{2} d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(2*c^2*d^2*g - (3*c^2*g*x^2 + 5*b*c*f - 2*b^2*g + (5*c^2*f + b*c*g)*x)*e^2 + (c^2*d*g*x + 5*c^2*d*f - 4*
b*c*d*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*sqrt(x*e + d)/(c^2*x*e^3 + c^2*d*e^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/sqrt(d + e*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (105) = 210\).
time = 1.32, size = 239, normalized size = 2.03 \begin {gather*} -\frac {2}{15} \, {\left (g {\left (\frac {2 \, \sqrt {2 \, c d - b e} c^{2} d^{2} + 3 \, \sqrt {2 \, c d - b e} b c d e - 2 \, \sqrt {2 \, c d - b e} b^{2} e^{2}}{c^{2}} + \frac {5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c d - 5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b e - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{c^{2}}\right )} e^{\left (-1\right )} + 5 \, f {\left (\frac {{\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}}}{c} - \frac {2 \, \sqrt {2 \, c d - b e} c d - \sqrt {2 \, c d - b e} b e}{c}\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(g*((2*sqrt(2*c*d - b*e)*c^2*d^2 + 3*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqrt(2*c*d - b*e)*b^2*e^2)/c^2 + (5*(
-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((x*e + d)*c - 2*c*d + b*
e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e))/c^2)*e^(-1) + 5*f*((-(x*e + d)*c + 2*c*d - b*e)^(3/2)/c - (2*sqrt(2*c*d
- b*e)*c*d - sqrt(2*c*d - b*e)*b*e)/c))*e^(-1)

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Mupad [B]
time = 2.50, size = 100, normalized size = 0.85 \begin {gather*} \frac {\left (\frac {2\,g\,x^2}{5}+\frac {2\,x\,\left (b\,e\,g-c\,d\,g+5\,c\,e\,f\right )}{15\,c\,e}+\frac {2\,\left (b\,e-c\,d\right )\,\left (2\,c\,d\,g-2\,b\,e\,g+5\,c\,e\,f\right )}{15\,c^2\,e^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((2*g*x^2)/5 + (2*x*(b*e*g - c*d*g + 5*c*e*f))/(15*c*e) + (2*(b*e - c*d)*(2*c*d*g - 2*b*e*g + 5*c*e*f))/(15*c
^2*e^2))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^(1/2)

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