3.5.100 \(\int \frac {(d+e x)^2}{a+c x^2} \, dx\) [500]

Optimal. Leaf size=59 \[ \frac {e^2 x}{c}+\frac {\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {d e \log \left (a+c x^2\right )}{c} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {716, 649, 211, 266} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (c d^2-a e^2\right )}{\sqrt {a} c^{3/2}}+\frac {d e \log \left (a+c x^2\right )}{c}+\frac {e^2 x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2/(a + c*x^2),x]

[Out]

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 716

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rubi steps

\begin {align*} \int \frac {(d+e x)^2}{a+c x^2} \, dx &=\int \left (\frac {e^2}{c}+\frac {c d^2-a e^2+2 c d e x}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac {e^2 x}{c}+\frac {\int \frac {c d^2-a e^2+2 c d e x}{a+c x^2} \, dx}{c}\\ &=\frac {e^2 x}{c}+(2 d e) \int \frac {x}{a+c x^2} \, dx+\frac {\left (c d^2-a e^2\right ) \int \frac {1}{a+c x^2} \, dx}{c}\\ &=\frac {e^2 x}{c}+\frac {\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {d e \log \left (a+c x^2\right )}{c}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 56, normalized size = 0.95 \begin {gather*} \frac {\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {e \left (e x+d \log \left (a+c x^2\right )\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2/(a + c*x^2),x]

[Out]

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

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Maple [A]
time = 0.45, size = 53, normalized size = 0.90

method	result	size
default	\(\frac {e^{2} x}{c}+\frac {d e \ln \left (c \,x^{2}+a \right )+\frac {\left (-e^{2} a +c \,d^{2}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{c}\)	\(53\)
risch	\(\frac {e^{2} x}{c}+\frac {\ln \left (-a^{2} e^{2}+a c \,d^{2}-\sqrt {-a c \left (e^{2} a -c \,d^{2}\right )^{2}}\, x \right ) d e}{c}+\frac {\ln \left (-a^{2} e^{2}+a c \,d^{2}-\sqrt {-a c \left (e^{2} a -c \,d^{2}\right )^{2}}\, x \right ) \sqrt {-a c \left (e^{2} a -c \,d^{2}\right )^{2}}}{2 c^{2} a}+\frac {\ln \left (-a^{2} e^{2}+a c \,d^{2}+\sqrt {-a c \left (e^{2} a -c \,d^{2}\right )^{2}}\, x \right ) d e}{c}-\frac {\ln \left (-a^{2} e^{2}+a c \,d^{2}+\sqrt {-a c \left (e^{2} a -c \,d^{2}\right )^{2}}\, x \right ) \sqrt {-a c \left (e^{2} a -c \,d^{2}\right )^{2}}}{2 c^{2} a}\)	\(232\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.54, size = 52, normalized size = 0.88 \begin {gather*} \frac {d e \log \left (c x^{2} + a\right )}{c} + \frac {x e^{2}}{c} + \frac {{\left (c d^{2} - a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*e*log(c*x^2 + a)/c + x*e^2/c + (c*d^2 - a*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*c)

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Fricas [A]
time = 2.83, size = 135, normalized size = 2.29 \begin {gather*} \left [\frac {2 \, a c d e \log \left (c x^{2} + a\right ) + 2 \, a c x e^{2} + {\left (c d^{2} - a e^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{2 \, a c^{2}}, \frac {a c d e \log \left (c x^{2} + a\right ) + a c x e^{2} + {\left (c d^{2} - a e^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{a c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*a*c*d*e*log(c*x^2 + a) + 2*a*c*x*e^2 + (c*d^2 - a*e^2)*sqrt(-a*c)*log((c*x^2 + 2*sqrt(-a*c)*x - a)/(c*
x^2 + a)))/(a*c^2), (a*c*d*e*log(c*x^2 + a) + a*c*x*e^2 + (c*d^2 - a*e^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a))/(a*
c^2)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (53) = 106\).
time = 0.24, size = 185, normalized size = 3.14 \begin {gather*} \left (\frac {d e}{c} - \frac {\sqrt {- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) \log {\left (x + \frac {- 2 a c \left (\frac {d e}{c} - \frac {\sqrt {- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) + 2 a d e}{a e^{2} - c d^{2}} \right )} + \left (\frac {d e}{c} + \frac {\sqrt {- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) \log {\left (x + \frac {- 2 a c \left (\frac {d e}{c} + \frac {\sqrt {- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) + 2 a d e}{a e^{2} - c d^{2}} \right )} + \frac {e^{2} x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.85, size = 52, normalized size = 0.88 \begin {gather*} \frac {d e \log \left (c x^{2} + a\right )}{c} + \frac {x e^{2}}{c} + \frac {{\left (c d^{2} - a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*e*log(c*x^2 + a)/c + x*e^2/c + (c*d^2 - a*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*c)

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Mupad [B]
time = 0.30, size = 62, normalized size = 1.05 \begin {gather*} \frac {e^2\,x}{c}+\frac {d^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {c}}-\frac {\sqrt {a}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{c^{3/2}}+\frac {d\,e\,\ln \left (c\,x^2+a\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^2/(a + c*x^2),x)

[Out]

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

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