3.5.64 \(\int \frac {(a+c x^2)^2}{d+e x} \, dx\) [464]

Optimal. Leaf size=94 \[ -\frac {c d \left (c d^2+2 a e^2\right ) x}{e^4}+\frac {c \left (c d^2+2 a e^2\right ) x^2}{2 e^3}-\frac {c^2 d x^3}{3 e^2}+\frac {c^2 x^4}{4 e}+\frac {\left (c d^2+a e^2\right )^2 \log (d+e x)}{e^5} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \begin {gather*} \frac {\left (a e^2+c d^2\right )^2 \log (d+e x)}{e^5}-\frac {c d x \left (2 a e^2+c d^2\right )}{e^4}+\frac {c x^2 \left (2 a e^2+c d^2\right )}{2 e^3}-\frac {c^2 d x^3}{3 e^2}+\frac {c^2 x^4}{4 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2/(d + e*x),x]

[Out]

-((c*d*(c*d^2 + 2*a*e^2)*x)/e^4) + (c*(c*d^2 + 2*a*e^2)*x^2)/(2*e^3) - (c^2*d*x^3)/(3*e^2) + (c^2*x^4)/(4*e) +
((c*d^2 + a*e^2)^2*Log[d + e*x])/e^5

Rule 711

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

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 79, normalized size = 0.84 \begin {gather*} \frac {c e x \left (12 a e^2 (-2 d+e x)+c \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+12 \left (c d^2+a e^2\right )^2 \log (d+e x)}{12 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^2/(d + e*x),x]

[Out]

(c*e*x*(12*a*e^2*(-2*d + e*x) + c*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 12*(c*d^2 + a*e^2)^2*Log[
d + e*x])/(12*e^5)

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Maple [A]
time = 0.48, size = 96, normalized size = 1.02

method	result	size
default	\(-\frac {c \left (-\frac {c \,x^{4} e^{3}}{4}+\frac {c d \,e^{2} x^{3}}{3}-\frac {\left (2 e^{2} a +c \,d^{2}\right ) x^{2} e}{2}+d \left (2 e^{2} a +c \,d^{2}\right ) x \right )}{e^{4}}+\frac {\left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(96\)
norman	\(\frac {c^{2} x^{4}}{4 e}+\frac {c \left (2 e^{2} a +c \,d^{2}\right ) x^{2}}{2 e^{3}}-\frac {c^{2} d \,x^{3}}{3 e^{2}}-\frac {c d \left (2 e^{2} a +c \,d^{2}\right ) x}{e^{4}}+\frac {\left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(101\)
risch	\(\frac {c^{2} x^{4}}{4 e}-\frac {c^{2} d \,x^{3}}{3 e^{2}}+\frac {c a \,x^{2}}{e}+\frac {c^{2} d^{2} x^{2}}{2 e^{3}}-\frac {2 c a d x}{e^{2}}-\frac {c^{2} d^{3} x}{e^{4}}+\frac {\ln \left (e x +d \right ) a^{2}}{e}+\frac {2 \ln \left (e x +d \right ) a c \,d^{2}}{e^{3}}+\frac {\ln \left (e x +d \right ) c^{2} d^{4}}{e^{5}}\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 99, normalized size = 1.05 \begin {gather*} {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-5\right )} \log \left (x e + d\right ) + \frac {1}{12} \, {\left (3 \, c^{2} x^{4} e^{3} - 4 \, c^{2} d x^{3} e^{2} + 6 \, {\left (c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{2} - 12 \, {\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*e^(-5)*log(x*e + d) + 1/12*(3*c^2*x^4*e^3 - 4*c^2*d*x^3*e^2 + 6*(c^2*d^2*e
+ 2*a*c*e^3)*x^2 - 12*(c^2*d^3 + 2*a*c*d*e^2)*x)*e^(-4)

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Fricas [A]
time = 2.64, size = 99, normalized size = 1.05 \begin {gather*} \frac {1}{12} \, {\left (6 \, c^{2} d^{2} x^{2} e^{2} - 12 \, c^{2} d^{3} x e + 3 \, {\left (c^{2} x^{4} + 4 \, a c x^{2}\right )} e^{4} - 4 \, {\left (c^{2} d x^{3} + 6 \, a c d x\right )} e^{3} + 12 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (x e + d\right )\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*c^2*d^2*x^2*e^2 - 12*c^2*d^3*x*e + 3*(c^2*x^4 + 4*a*c*x^2)*e^4 - 4*(c^2*d*x^3 + 6*a*c*d*x)*e^3 + 12*(c
^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*log(x*e + d))*e^(-5)

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Sympy [A]
time = 0.12, size = 88, normalized size = 0.94 \begin {gather*} - \frac {c^{2} d x^{3}}{3 e^{2}} + \frac {c^{2} x^{4}}{4 e} + x^{2} \left (\frac {a c}{e} + \frac {c^{2} d^{2}}{2 e^{3}}\right ) + x \left (- \frac {2 a c d}{e^{2}} - \frac {c^{2} d^{3}}{e^{4}}\right ) + \frac {\left (a e^{2} + c d^{2}\right )^{2} \log {\left (d + e x \right )}}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.02, size = 100, normalized size = 1.06 \begin {gather*} {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, c^{2} x^{4} e^{3} - 4 \, c^{2} d x^{3} e^{2} + 6 \, c^{2} d^{2} x^{2} e - 12 \, c^{2} d^{3} x + 12 \, a c x^{2} e^{3} - 24 \, a c d x e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*e^(-5)*log(abs(x*e + d)) + 1/12*(3*c^2*x^4*e^3 - 4*c^2*d*x^3*e^2 + 6*c^2*d
^2*x^2*e - 12*c^2*d^3*x + 12*a*c*x^2*e^3 - 24*a*c*d*x*e^2)*e^(-4)

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Mupad [B]
time = 0.04, size = 106, normalized size = 1.13 \begin {gather*} x^2\,\left (\frac {c^2\,d^2}{2\,e^3}+\frac {a\,c}{e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{e^5}+\frac {c^2\,x^4}{4\,e}-\frac {c^2\,d\,x^3}{3\,e^2}-\frac {d\,x\,\left (\frac {c^2\,d^2}{e^3}+\frac {2\,a\,c}{e}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2/(d + e*x),x)

[Out]

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

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