\(\int \frac {(d+e x)^2}{(a+c x^2)^{5/2}} \, dx\) [580]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {x (d+e x)^2}{3 a \left (a+c x^2\right )^{3/2}}-\frac {2 d (a e-c d x)}{3 a^2 c \sqrt {a+c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {743, 651} \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {x (d+e x)^2}{3 a \left (a+c x^2\right )^{3/2}}-\frac {2 d (a e-c d x)}{3 a^2 c \sqrt {a+c x^2}} \]

[In]

Int[(d + e*x)^2/(a + c*x^2)^(5/2),x]

[Out]

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

Rule 651

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((-a)*e + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rule 743

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^m)*(2*c*x)*((a + c*x^2)
^(p + 1)/(4*a*c*(p + 1))), x] - Dist[m*((2*c*d)/(4*a*c*(p + 1))), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x
], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]

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

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {-2 a^2 d e+3 a c d^2 x+2 c^2 d^2 x^3+a c e^2 x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}} \]

[In]

Integrate[(d + e*x)^2/(a + c*x^2)^(5/2),x]

[Out]

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

Maple [A] (verified)

Time = 2.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95

method	result	size
gosper	\(-\frac {-a c \,e^{2} x^{3}-2 c^{2} d^{2} x^{3}-3 a c \,d^{2} x +2 a^{2} d e}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c}\)	\(55\)
trager	\(-\frac {-a c \,e^{2} x^{3}-2 c^{2} d^{2} x^{3}-3 a c \,d^{2} x +2 a^{2} d e}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c}\)	\(55\)
default	\(d^{2} \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )+e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )}{2 c}\right )-\frac {2 d e}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(110\)

[In]

int((e*x+d)^2/(c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {{\left (3 \, a c d^{2} x - 2 \, a^{2} d e + {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{3}\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \]

[In]

integrate((e*x+d)^2/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (a + c x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((e*x+d)**2/(c*x**2+a)**(5/2),x)

[Out]

Integral((d + e*x)**2/(a + c*x**2)**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.59 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {2 \, d^{2} x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {d^{2} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {e^{2} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {e^{2} x}{3 \, \sqrt {c x^{2} + a} a c} - \frac {2 \, d e}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} \]

[In]

integrate((e*x+d)^2/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {{\left (\frac {3 \, d^{2}}{a} + \frac {{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}}{a^{2} c}\right )} x - \frac {2 \, d e}{c}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \]

[In]

integrate((e*x+d)^2/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.35 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {a\,e^2\,x\,\left (c\,x^2+a\right )-2\,a^2\,d\,e-a^2\,e^2\,x+2\,c\,d^2\,x\,\left (c\,x^2+a\right )+a\,c\,d^2\,x}{3\,a^2\,c\,{\left (c\,x^2+a\right )}^{3/2}} \]

[In]

int((d + e*x)^2/(a + c*x^2)^(5/2),x)

[Out]

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