3.6.13 \(\int \frac {(d+e x)^4}{(a+c x^2)^3} \, dx\) [513]

Optimal. Leaf size=120 \[ -\frac {(a e-c d x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac {3 \left (c d^2+a e^2\right ) (a e-c d x) (d+e x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {3 \left (c d^2+a e^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}} \]

[Out]

-1/4*(-c*d*x+a*e)*(e*x+d)^3/a/c/(c*x^2+a)^2-3/8*(a*e^2+c*d^2)*(-c*d*x+a*e)*(e*x+d)/a^2/c^2/(c*x^2+a)+3/8*(a*e^
2+c*d^2)^2*arctan(x*c^(1/2)/a^(1/2))/a^(5/2)/c^(5/2)

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Rubi [A]
time = 0.03, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {737, 211} \begin {gather*} \frac {3 \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a e^2+c d^2\right )^2}{8 a^{5/2} c^{5/2}}-\frac {3 (d+e x) \left (a e^2+c d^2\right ) (a e-c d x)}{8 a^2 c^2 \left (a+c x^2\right )}-\frac {(d+e x)^3 (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4/(a + c*x^2)^3,x]

[Out]

-1/4*((a*e - c*d*x)*(d + e*x)^3)/(a*c*(a + c*x^2)^2) - (3*(c*d^2 + a*e^2)*(a*e - c*d*x)*(d + e*x))/(8*a^2*c^2*
(a + c*x^2)) + (3*(c*d^2 + a*e^2)^2*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(5/2))

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 737

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a*e - c*d*x)*((a
+ c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Dist[(2*p + 3)*((c*d^2 + a*e^2)/(2*a*c*(p + 1))), Int[(d + e*x)^(m -
2)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && Lt
Q[p, -1]

Rubi steps

\begin {align*} \int \frac {(d+e x)^4}{\left (a+c x^2\right )^3} \, dx &=-\frac {(a e-c d x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}+\frac {\left (3 \left (c d^2+a e^2\right )\right ) \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {(a e-c d x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac {3 \left (c d^2+a e^2\right ) (a e-c d x) (d+e x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (3 \left (c d^2+a e^2\right )^2\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac {(a e-c d x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac {3 \left (c d^2+a e^2\right ) (a e-c d x) (d+e x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {3 \left (c d^2+a e^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 148, normalized size = 1.23 \begin {gather*} \frac {3 c^3 d^4 x^3-a^3 e^3 (8 d+3 e x)+a c^2 d^2 x \left (5 d^2+6 e^2 x^2\right )-a^2 c e \left (8 d^3+6 d^2 e x+16 d e^2 x^2+5 e^3 x^3\right )}{8 a^2 c^2 \left (a+c x^2\right )^2}+\frac {3 \left (c d^2+a e^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4/(a + c*x^2)^3,x]

[Out]

(3*c^3*d^4*x^3 - a^3*e^3*(8*d + 3*e*x) + a*c^2*d^2*x*(5*d^2 + 6*e^2*x^2) - a^2*c*e*(8*d^3 + 6*d^2*e*x + 16*d*e
^2*x^2 + 5*e^3*x^3))/(8*a^2*c^2*(a + c*x^2)^2) + (3*(c*d^2 + a*e^2)^2*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*
c^(5/2))

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Maple [A]
time = 0.47, size = 164, normalized size = 1.37

method	result	size
default	\(\frac {-\frac {\left (5 a^{2} e^{4}-6 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right ) x^{3}}{8 a^{2} c}-\frac {2 d \,e^{3} x^{2}}{c}-\frac {\left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}-5 c^{2} d^{4}\right ) x}{8 c^{2} a}-\frac {d e \left (e^{2} a +c \,d^{2}\right )}{c^{2}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {3 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 a^{2} c^{2} \sqrt {a c}}\)	\(164\)
risch	\(\frac {-\frac {\left (5 a^{2} e^{4}-6 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right ) x^{3}}{8 a^{2} c}-\frac {2 d \,e^{3} x^{2}}{c}-\frac {\left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}-5 c^{2} d^{4}\right ) x}{8 c^{2} a}-\frac {d e \left (e^{2} a +c \,d^{2}\right )}{c^{2}}}{\left (c \,x^{2}+a \right )^{2}}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) e^{4}}{16 \sqrt {-a c}\, c^{2}}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) d^{2} e^{2}}{8 \sqrt {-a c}\, c a}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) d^{4}}{16 \sqrt {-a c}\, a^{2}}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) e^{4}}{16 \sqrt {-a c}\, c^{2}}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) d^{2} e^{2}}{8 \sqrt {-a c}\, c a}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) d^{4}}{16 \sqrt {-a c}\, a^{2}}\)	\(282\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(c*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

(-1/8*(5*a^2*e^4-6*a*c*d^2*e^2-3*c^2*d^4)/a^2/c*x^3-2*d*e^3*x^2/c-1/8*(3*a^2*e^4+6*a*c*d^2*e^2-5*c^2*d^4)/c^2/
a*x-d*e*(a*e^2+c*d^2)/c^2)/(c*x^2+a)^2+3/8*(a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)/a^2/c^2/(a*c)^(1/2)*arctan(c*x/(a*c
)^(1/2))

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Maxima [A]
time = 0.55, size = 175, normalized size = 1.46 \begin {gather*} -\frac {16 \, a^{2} c d x^{2} e^{3} + 8 \, a^{2} c d^{3} e + 8 \, a^{3} d e^{3} - {\left (3 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} - 5 \, a^{2} c e^{4}\right )} x^{3} - {\left (5 \, a c^{2} d^{4} - 6 \, a^{2} c d^{2} e^{2} - 3 \, a^{3} e^{4}\right )} x}{8 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} + \frac {3 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/8*(16*a^2*c*d*x^2*e^3 + 8*a^2*c*d^3*e + 8*a^3*d*e^3 - (3*c^3*d^4 + 6*a*c^2*d^2*e^2 - 5*a^2*c*e^4)*x^3 - (5*
a*c^2*d^4 - 6*a^2*c*d^2*e^2 - 3*a^3*e^4)*x)/(a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2) + 3/8*(c^2*d^4 + 2*a*c*d^2
*e^2 + a^2*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (108) = 216\).
time = 1.73, size = 544, normalized size = 4.53 \begin {gather*} \left [\frac {6 \, a c^{4} d^{4} x^{3} + 10 \, a^{2} c^{3} d^{4} x - 16 \, a^{3} c^{2} d^{3} e - 3 \, {\left (c^{4} d^{4} x^{4} + 2 \, a c^{3} d^{4} x^{2} + a^{2} c^{2} d^{4} + {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} e^{4} + 2 \, {\left (a c^{3} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{3} c d^{2}\right )} 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 (5 \, a^{3} c^{2} x^{3} + 3 \, a^{4} c x\right )} e^{4} - 16 \, {\left (2 \, a^{3} c^{2} d x^{2} + a^{4} c d\right )} e^{3} + 12 \, {\left (a^{2} c^{3} d^{2} x^{3} - a^{3} c^{2} d^{2} x\right )} e^{2}}{16 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, \frac {3 \, a c^{4} d^{4} x^{3} + 5 \, a^{2} c^{3} d^{4} x - 8 \, a^{3} c^{2} d^{3} e + 3 \, {\left (c^{4} d^{4} x^{4} + 2 \, a c^{3} d^{4} x^{2} + a^{2} c^{2} d^{4} + {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} e^{4} + 2 \, {\left (a c^{3} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{3} c d^{2}\right )} e^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (5 \, a^{3} c^{2} x^{3} + 3 \, a^{4} c x\right )} e^{4} - 8 \, {\left (2 \, a^{3} c^{2} d x^{2} + a^{4} c d\right )} e^{3} + 6 \, {\left (a^{2} c^{3} d^{2} x^{3} - a^{3} c^{2} d^{2} x\right )} e^{2}}{8 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/16*(6*a*c^4*d^4*x^3 + 10*a^2*c^3*d^4*x - 16*a^3*c^2*d^3*e - 3*(c^4*d^4*x^4 + 2*a*c^3*d^4*x^2 + a^2*c^2*d^4
+ (a^2*c^2*x^4 + 2*a^3*c*x^2 + a^4)*e^4 + 2*(a*c^3*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^3*c*d^2)*e^2)*sqrt(-a*c)*lo
g((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) - 2*(5*a^3*c^2*x^3 + 3*a^4*c*x)*e^4 - 16*(2*a^3*c^2*d*x^2 + a^4*c*
d)*e^3 + 12*(a^2*c^3*d^2*x^3 - a^3*c^2*d^2*x)*e^2)/(a^3*c^5*x^4 + 2*a^4*c^4*x^2 + a^5*c^3), 1/8*(3*a*c^4*d^4*x
^3 + 5*a^2*c^3*d^4*x - 8*a^3*c^2*d^3*e + 3*(c^4*d^4*x^4 + 2*a*c^3*d^4*x^2 + a^2*c^2*d^4 + (a^2*c^2*x^4 + 2*a^3
*c*x^2 + a^4)*e^4 + 2*(a*c^3*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^3*c*d^2)*e^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) - (
5*a^3*c^2*x^3 + 3*a^4*c*x)*e^4 - 8*(2*a^3*c^2*d*x^2 + a^4*c*d)*e^3 + 6*(a^2*c^3*d^2*x^3 - a^3*c^2*d^2*x)*e^2)/
(a^3*c^5*x^4 + 2*a^4*c^4*x^2 + a^5*c^3)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (112) = 224\).
time = 1.14, size = 328, normalized size = 2.73 \begin {gather*} - \frac {3 \sqrt {- \frac {1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2} \log {\left (- \frac {3 a^{3} c^{2} \sqrt {- \frac {1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2}}{3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 3 c^{2} d^{4}} + x \right )}}{16} + \frac {3 \sqrt {- \frac {1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2} \log {\left (\frac {3 a^{3} c^{2} \sqrt {- \frac {1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2}}{3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 3 c^{2} d^{4}} + x \right )}}{16} + \frac {- 8 a^{3} d e^{3} - 8 a^{2} c d^{3} e - 16 a^{2} c d e^{3} x^{2} + x^{3} \left (- 5 a^{2} c e^{4} + 6 a c^{2} d^{2} e^{2} + 3 c^{3} d^{4}\right ) + x \left (- 3 a^{3} e^{4} - 6 a^{2} c d^{2} e^{2} + 5 a c^{2} d^{4}\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(c*x**2+a)**3,x)

[Out]

-3*sqrt(-1/(a**5*c**5))*(a*e**2 + c*d**2)**2*log(-3*a**3*c**2*sqrt(-1/(a**5*c**5))*(a*e**2 + c*d**2)**2/(3*a**
2*e**4 + 6*a*c*d**2*e**2 + 3*c**2*d**4) + x)/16 + 3*sqrt(-1/(a**5*c**5))*(a*e**2 + c*d**2)**2*log(3*a**3*c**2*
sqrt(-1/(a**5*c**5))*(a*e**2 + c*d**2)**2/(3*a**2*e**4 + 6*a*c*d**2*e**2 + 3*c**2*d**4) + x)/16 + (-8*a**3*d*e
**3 - 8*a**2*c*d**3*e - 16*a**2*c*d*e**3*x**2 + x**3*(-5*a**2*c*e**4 + 6*a*c**2*d**2*e**2 + 3*c**3*d**4) + x*(
-3*a**3*e**4 - 6*a**2*c*d**2*e**2 + 5*a*c**2*d**4))/(8*a**4*c**2 + 16*a**3*c**3*x**2 + 8*a**2*c**4*x**4)

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Giac [A]
time = 1.64, size = 161, normalized size = 1.34 \begin {gather*} \frac {3 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} + \frac {3 \, c^{3} d^{4} x^{3} + 6 \, a c^{2} d^{2} x^{3} e^{2} + 5 \, a c^{2} d^{4} x - 5 \, a^{2} c x^{3} e^{4} - 16 \, a^{2} c d x^{2} e^{3} - 6 \, a^{2} c d^{2} x e^{2} - 8 \, a^{2} c d^{3} e - 3 \, a^{3} x e^{4} - 8 \, a^{3} d e^{3}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+a)^3,x, algorithm="giac")

[Out]

3/8*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c^2) + 1/8*(3*c^3*d^4*x^3 + 6*a*c
^2*d^2*x^3*e^2 + 5*a*c^2*d^4*x - 5*a^2*c*x^3*e^4 - 16*a^2*c*d*x^2*e^3 - 6*a^2*c*d^2*x*e^2 - 8*a^2*c*d^3*e - 3*
a^3*x*e^4 - 8*a^3*d*e^3)/((c*x^2 + a)^2*a^2*c^2)

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Mupad [B]
time = 0.16, size = 197, normalized size = 1.64 \begin {gather*} \frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x\,{\left (c\,d^2+a\,e^2\right )}^2}{\sqrt {a}\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}\right )\,{\left (c\,d^2+a\,e^2\right )}^2}{8\,a^{5/2}\,c^{5/2}}-\frac {\frac {2\,d\,e^3\,x^2}{c}+\frac {x\,\left (3\,a^2\,e^4+6\,a\,c\,d^2\,e^2-5\,c^2\,d^4\right )}{8\,a\,c^2}+\frac {d\,e\,\left (c\,d^2+a\,e^2\right )}{c^2}-\frac {x^3\,\left (-5\,a^2\,e^4+6\,a\,c\,d^2\,e^2+3\,c^2\,d^4\right )}{8\,a^2\,c}}{a^2+2\,a\,c\,x^2+c^2\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4/(a + c*x^2)^3,x)

[Out]

(3*atan((c^(1/2)*x*(a*e^2 + c*d^2)^2)/(a^(1/2)*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)))*(a*e^2 + c*d^2)^2)/(8*a^(
5/2)*c^(5/2)) - ((2*d*e^3*x^2)/c + (x*(3*a^2*e^4 - 5*c^2*d^4 + 6*a*c*d^2*e^2))/(8*a*c^2) + (d*e*(a*e^2 + c*d^2
))/c^2 - (x^3*(3*c^2*d^4 - 5*a^2*e^4 + 6*a*c*d^2*e^2))/(8*a^2*c))/(a^2 + c^2*x^4 + 2*a*c*x^2)

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