3.5.19 \(\int \frac {(d+e x)^3}{a+c x^2} \, dx\)

Optimal. Leaf size=90 \[ \frac {d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}+\frac {3 d e^2 x}{c}+\frac {e^3 x^2}{2 c} \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 90, 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 = {702, 635, 205, 260} \begin {gather*} \frac {e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}+\frac {d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {3 d e^2 x}{c}+\frac {e^3 x^2}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3/(a + c*x^2),x]

[Out]

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

Rule 205

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

Rule 260

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 635

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 702

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

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 80, normalized size = 0.89 \begin {gather*} \frac {d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {e \left (\left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )+c e x (6 d+e x)\right )}{2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3/(a + c*x^2),x]

[Out]

(d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (e*(c*e*x*(6*d + e*x) + (3*c*d^2 - a*e^2
)*Log[a + c*x^2]))/(2*c^2)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^3}{a+c x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^3/(a + c*x^2),x]

[Out]

IntegrateAlgebraic[(d + e*x)^3/(a + c*x^2), x]

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 187, normalized size = 2.08 \begin {gather*} \left [\frac {a c e^{3} x^{2} + 6 \, a c d e^{2} x + {\left (c d^{3} - 3 \, a d e^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + {\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}, \frac {a c e^{3} x^{2} + 6 \, a c d e^{2} x + 2 \, {\left (c d^{3} - 3 \, a d e^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+a),x, algorithm="fricas")

[Out]

[1/2*(a*c*e^3*x^2 + 6*a*c*d*e^2*x + (c*d^3 - 3*a*d*e^2)*sqrt(-a*c)*log((c*x^2 + 2*sqrt(-a*c)*x - a)/(c*x^2 + a
)) + (3*a*c*d^2*e - a^2*e^3)*log(c*x^2 + a))/(a*c^2), 1/2*(a*c*e^3*x^2 + 6*a*c*d*e^2*x + 2*(c*d^3 - 3*a*d*e^2)
*sqrt(a*c)*arctan(sqrt(a*c)*x/a) + (3*a*c*d^2*e - a^2*e^3)*log(c*x^2 + a))/(a*c^2)]

________________________________________________________________________________________

giac [A]  time = 0.15, size = 78, normalized size = 0.87 \begin {gather*} \frac {{\left (c d^{3} - 3 \, a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} + \frac {{\left (3 \, c d^{2} e - a e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac {c x^{2} e^{3} + 6 \, c d x e^{2}}{2 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+a),x, algorithm="giac")

[Out]

(c*d^3 - 3*a*d*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*c) + 1/2*(3*c*d^2*e - a*e^3)*log(c*x^2 + a)/c^2 + 1/2*(c*
x^2*e^3 + 6*c*d*x*e^2)/c^2

________________________________________________________________________________________

maple [A]  time = 0.04, size = 99, normalized size = 1.10 \begin {gather*} -\frac {3 a d \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c}+\frac {e^{3} x^{2}}{2 c}+\frac {d^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}-\frac {a \,e^{3} \ln \left (c \,x^{2}+a \right )}{2 c^{2}}+\frac {3 d^{2} e \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {3 d \,e^{2} x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 3.02, size = 79, normalized size = 0.88 \begin {gather*} \frac {{\left (c d^{3} - 3 \, a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} + \frac {e^{3} x^{2} + 6 \, d e^{2} x}{2 \, c} + \frac {{\left (3 \, c d^{2} e - a e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+a),x, algorithm="maxima")

[Out]

(c*d^3 - 3*a*d*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*c) + 1/2*(e^3*x^2 + 6*d*e^2*x)/c + 1/2*(3*c*d^2*e - a*e^3
)*log(c*x^2 + a)/c^2

________________________________________________________________________________________

mupad [B]  time = 0.13, size = 91, normalized size = 1.01 \begin {gather*} \frac {e^3\,x^2}{2\,c}-\frac {\ln \left (c\,x^2+a\right )\,\left (4\,a^2\,c^2\,e^3-12\,a\,c^3\,d^2\,e\right )}{8\,a\,c^4}+\frac {3\,d\,e^2\,x}{c}-\frac {d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (3\,a\,e^2-c\,d^2\right )}{\sqrt {a}\,c^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^3/(a + c*x^2),x)

[Out]

(e^3*x^2)/(2*c) - (log(a + c*x^2)*(4*a^2*c^2*e^3 - 12*a*c^3*d^2*e))/(8*a*c^4) + (3*d*e^2*x)/c - (d*atan((c^(1/
2)*x)/a^(1/2))*(3*a*e^2 - c*d^2))/(a^(1/2)*c^(3/2))

________________________________________________________________________________________

sympy [B]  time = 0.80, size = 308, normalized size = 3.42 \begin {gather*} \left (- \frac {e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} - \frac {d \sqrt {- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) \log {\left (x + \frac {- a^{2} e^{3} - 2 a c^{2} \left (- \frac {e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} - \frac {d \sqrt {- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) + 3 a c d^{2} e}{3 a c d e^{2} - c^{2} d^{3}} \right )} + \left (- \frac {e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} + \frac {d \sqrt {- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) \log {\left (x + \frac {- a^{2} e^{3} - 2 a c^{2} \left (- \frac {e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} + \frac {d \sqrt {- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) + 3 a c d^{2} e}{3 a c d e^{2} - c^{2} d^{3}} \right )} + \frac {3 d e^{2} x}{c} + \frac {e^{3} x^{2}}{2 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(c*x**2+a),x)

[Out]

(-e*(a*e**2 - 3*c*d**2)/(2*c**2) - d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4))*log(x + (-a**2*e**3 - 2*a*c
**2*(-e*(a*e**2 - 3*c*d**2)/(2*c**2) - d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4)) + 3*a*c*d**2*e)/(3*a*c*
d*e**2 - c**2*d**3)) + (-e*(a*e**2 - 3*c*d**2)/(2*c**2) + d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4))*log(
x + (-a**2*e**3 - 2*a*c**2*(-e*(a*e**2 - 3*c*d**2)/(2*c**2) + d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4))
+ 3*a*c*d**2*e)/(3*a*c*d*e**2 - c**2*d**3)) + 3*d*e**2*x/c + e**3*x**2/(2*c)

________________________________________________________________________________________