3.21.14 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=118 \[ -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g-3 c d g+7 c e f)}{35 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}} \]

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Rubi [A]  time = 0.19, 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 = {794, 648} \begin {gather*} -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g-3 c d g+7 c e f)}{35 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(-2*(7*c*e*f - 3*c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(35*c^2*e^2*(d + e*x)^(5/2)) -
(2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(7*c*e^2*(d + e*x)^(3/2))

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 {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}}-\frac {\left (2 \left (\frac {5}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac {3}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{7 c e^3}\\ &=-\frac {2 (7 c e f-3 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 78, normalized size = 0.66 \begin {gather*} -\frac {2 (b e-c d+c e x)^2 \sqrt {(d+e x) (c (d-e x)-b e)} (c (2 d g+7 e f+5 e g x)-2 b e g)}{35 c^2 e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-2*b*e*g + c*(7*e*f + 2*d*g + 5*e*g*x)))/
(35*c^2*e^2*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 1.59, size = 74, normalized size = 0.63 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{5/2} (-2 b e g+5 c g (d+e x)-3 c d g+7 c e f)}{35 c^2 e^2 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(-2*(7*c*e*f - 3*c*d*g - 2*b*e*g + 5*c*g*(d + e*x))*((2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2)^(5/2))/(35*c^2*e
^2*(d + e*x)^(5/2))

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fricas [B]  time = 0.40, size = 229, normalized size = 1.94 \begin {gather*} -\frac {2 \, {\left (5 \, c^{3} e^{3} g x^{3} + {\left (7 \, c^{3} e^{3} f - 8 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + 2 \, {\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} g - {\left (14 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} f - {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{35 \, {\left (c^{2} e^{3} x + 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)^(3/2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

-2/35*(5*c^3*e^3*g*x^3 + (7*c^3*e^3*f - 8*(c^3*d*e^2 - b*c^2*e^3)*g)*x^2 + 7*(c^3*d^2*e - 2*b*c^2*d*e^2 + b^2*
c*e^3)*f + 2*(c^3*d^3 - 3*b*c^2*d^2*e + 3*b^2*c*d*e^2 - b^3*e^3)*g - (14*(c^3*d*e^2 - b*c^2*e^3)*f - (c^3*d^2*
e - 2*b*c^2*d*e^2 + b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^2*e^3*x + c^2
*d*e^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \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)^(3/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 79, normalized size = 0.67 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-5 c e g x +2 b e g -2 c d g -7 c e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{\frac {3}{2}} c^{2} e^{2}} \end {gather*}

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)^(3/2)/(e*x+d)^(3/2),x)

[Out]

-2/35*(c*e*x+b*e-c*d)*(-5*c*e*g*x+2*b*e*g-2*c*d*g-7*c*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c^2/e^2/(e*x
+d)^(3/2)

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maxima [A]  time = 0.75, size = 197, normalized size = 1.67 \begin {gather*} -\frac {2 \, {\left (c^{2} e^{2} x^{2} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x\right )} \sqrt {-c e x + c d - b e} f}{5 \, c e} - \frac {2 \, {\left (5 \, c^{3} e^{3} x^{3} + 2 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3} - 8 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} + {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{35 \, c^{2} e^{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)^(3/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

-2/5*(c^2*e^2*x^2 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*(c^2*d*e - b*c*e^2)*x)*sqrt(-c*e*x + c*d - b*e)*f/(c*e)
- 2/35*(5*c^3*e^3*x^3 + 2*c^3*d^3 - 6*b*c^2*d^2*e + 6*b^2*c*d*e^2 - 2*b^3*e^3 - 8*(c^3*d*e^2 - b*c^2*e^3)*x^2
+ (c^3*d^2*e - 2*b*c^2*d*e^2 + b^2*c*e^3)*x)*sqrt(-c*e*x + c*d - b*e)*g/(c^2*e^2)

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mupad [B]  time = 2.81, size = 133, normalized size = 1.13 \begin {gather*} -\frac {\left (x^2\,\left (\frac {16\,b\,e\,g}{35}-\frac {16\,c\,d\,g}{35}+\frac {2\,c\,e\,f}{5}\right )+\frac {2\,c\,e\,g\,x^3}{7}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\left (2\,c\,d\,g-2\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^2\,e^2}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\left (b\,e\,g-c\,d\,g+14\,c\,e\,f\right )}{35\,c\,e}\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)^(3/2))/(d + e*x)^(3/2),x)

[Out]

-((x^2*((16*b*e*g)/35 - (16*c*d*g)/35 + (2*c*e*f)/5) + (2*c*e*g*x^3)/7 + (2*(b*e - c*d)^2*(2*c*d*g - 2*b*e*g +
7*c*e*f))/(35*c^2*e^2) + (2*x*(b*e - c*d)*(b*e*g - c*d*g + 14*c*e*f))/(35*c*e))*(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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, 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)**(3/2)/(e*x+d)**(3/2),x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**(3/2), x)

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