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

Optimal. Leaf size=58 \[ \frac {x (d+e x)^2}{3 a \left (a+c x^2\right )^{3/2}}-\frac {2 d (a e-c d x)}{3 a^2 c \sqrt {a+c x^2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {729, 637} \begin {gather*} \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 {gather*}

Antiderivative was successfully verified.

[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 637

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 729

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*} \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) \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*}

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Mathematica [A]  time = 0.08, size = 57, normalized size = 0.98 \begin {gather*} \frac {-2 a^2 d e+a c x \left (3 d^2+e^2 x^2\right )+2 c^2 d^2 x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.55, size = 57, normalized size = 0.98 \begin {gather*} \frac {-2 a^2 d e+3 a c d^2 x+a c e^2 x^3+2 c^2 d^2 x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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))

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fricas [A]  time = 0.42, size = 75, normalized size = 1.29 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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
)

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giac [A]  time = 0.24, size = 55, normalized size = 0.95 \begin {gather*} \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}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.05, size = 55, normalized size = 0.95 \begin {gather*} -\frac {-a c \,e^{2} x^{3}-2 c^{2} d^{2} x^{3}-3 d^{2} x a c +2 a^{2} d e}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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maxima [A]  time = 1.32, size = 92, normalized size = 1.59 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.44, size = 68, normalized size = 1.17 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{2}}{\left (a + c x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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