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

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

[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)

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Rubi [A]
time = 0.01, 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 = {743, 651} \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 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*} \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 = 0.45, size = 57, normalized size = 0.98 \begin {gather*} \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}} \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 + 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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(109\) vs. \(2(50)=100\).
time = 0.45, size = 110, normalized size = 1.90

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 91, normalized size = 1.57 \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 {x e^{2}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {x e^{2}}{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*x*e^2/((c*x^2 + a)^(3/2)*c) + 1/3*x*e^
2/(sqrt(c*x^2 + a)*a*c) - 2/3*d*e/((c*x^2 + a)^(3/2)*c)

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Fricas [A]
time = 4.76, size = 76, normalized size = 1.31 \begin {gather*} \frac {{\left (2 \, c^{2} d^{2} x^{3} + a c x^{3} e^{2} + 3 \, a c d^{2} x - 2 \, a^{2} d e\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*(2*c^2*d^2*x^3 + a*c*x^3*e^2 + 3*a*c*d^2*x - 2*a^2*d*e)*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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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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Giac [A]
time = 3.32, 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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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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