3.1332 \(\int \frac{(A+B x) (d+e x)^2}{a+c x^2} \, dx\)

Optimal. Leaf size=108 \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+B c d^2\right )}{2 c^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (-a A e^2-2 a B d e+A c d^2\right )}{\sqrt{a} c^{3/2}}+\frac{e x (A e+2 B d)}{c}+\frac{B e^2 x^2}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.0946352, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {801, 635, 205, 260} \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+B c d^2\right )}{2 c^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (-a A e^2-2 a B d e+A c d^2\right )}{\sqrt{a} c^{3/2}}+\frac{e x (A e+2 B d)}{c}+\frac{B e^2 x^2}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^2)/(a + c*x^2),x]

[Out]

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

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0970936, size = 99, normalized size = 0.92 \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+B c d^2\right )-\frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a A e^2+2 a B d e-A c d^2\right )}{\sqrt{a}}+c e x (2 A e+4 B d+B e x)}{2 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*e*x*(4*B*d + 2*A*e + B*e*x) - (2*Sqrt[c]*(-(A*c*d^2) + 2*a*B*d*e + a*A*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/Sq
rt[a] + (B*c*d^2 + 2*A*c*d*e - a*B*e^2)*Log[a + c*x^2])/(2*c^2)

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Maple [A]  time = 0.006, size = 148, normalized size = 1.4 \begin{align*}{\frac{B{e}^{2}{x}^{2}}{2\,c}}+{\frac{A{e}^{2}x}{c}}+2\,{\frac{Bdex}{c}}+{\frac{\ln \left ( c{x}^{2}+a \right ) Ade}{c}}-{\frac{\ln \left ( c{x}^{2}+a \right ) aB{e}^{2}}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+a \right ) B{d}^{2}}{2\,c}}-{\frac{Aa{e}^{2}}{c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{A{d}^{2}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-2\,{\frac{aBde}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.97686, size = 533, normalized size = 4.94 \begin{align*} \left [\frac{B a c e^{2} x^{2} +{\left (A c d^{2} - 2 \, B a d e - A 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 \,{\left (2 \, B a c d e + A a c e^{2}\right )} x +{\left (B a c d^{2} + 2 \, A a c d e - B a^{2} e^{2}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}, \frac{B a c e^{2} x^{2} + 2 \,{\left (A c d^{2} - 2 \, B a d e - A a e^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) + 2 \,{\left (2 \, B a c d e + A a c e^{2}\right )} x +{\left (B a c d^{2} + 2 \, A a c d e - B a^{2} e^{2}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 3.14249, size = 423, normalized size = 3.92 \begin{align*} \frac{B e^{2} x^{2}}{2 c} + \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2}}{2 c^{2}} - \frac{\sqrt{- a c^{5}} \left (A a e^{2} - A c d^{2} + 2 B a d e\right )}{2 a c^{4}}\right ) \log{\left (x + \frac{2 A a c d e - B a^{2} e^{2} + B a c d^{2} - 2 a c^{2} \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2}}{2 c^{2}} - \frac{\sqrt{- a c^{5}} \left (A a e^{2} - A c d^{2} + 2 B a d e\right )}{2 a c^{4}}\right )}{A a c e^{2} - A c^{2} d^{2} + 2 B a c d e} \right )} + \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2}}{2 c^{2}} + \frac{\sqrt{- a c^{5}} \left (A a e^{2} - A c d^{2} + 2 B a d e\right )}{2 a c^{4}}\right ) \log{\left (x + \frac{2 A a c d e - B a^{2} e^{2} + B a c d^{2} - 2 a c^{2} \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2}}{2 c^{2}} + \frac{\sqrt{- a c^{5}} \left (A a e^{2} - A c d^{2} + 2 B a d e\right )}{2 a c^{4}}\right )}{A a c e^{2} - A c^{2} d^{2} + 2 B a c d e} \right )} + \frac{x \left (A e^{2} + 2 B d e\right )}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13471, size = 136, normalized size = 1.26 \begin{align*} \frac{{\left (A c d^{2} - 2 \, B a d e - A a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c} + \frac{{\left (B c d^{2} + 2 \, A c d e - B a e^{2}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{B c x^{2} e^{2} + 4 \, B c d x e + 2 \, A c x e^{2}}{2 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*c*d^2 - 2*B*a*d*e - A*a*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*c) + 1/2*(B*c*d^2 + 2*A*c*d*e - B*a*e^2)*log(
c*x^2 + a)/c^2 + 1/2*(B*c*x^2*e^2 + 4*B*c*d*x*e + 2*A*c*x*e^2)/c^2