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

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

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Rubi [A]  time = 0.08, 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 = {697} \begin {gather*} \frac {c x^2 \left (2 a e^2+c d^2\right )}{2 e^3}-\frac {c d x \left (2 a e^2+c d^2\right )}{e^4}+\frac {\left (a e^2+c d^2\right )^2 \log (d+e x)}{e^5}-\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 697

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 = 0.03, size = 79, normalized size = 0.84 \begin {gather*} \frac {12 \left (a e^2+c d^2\right )^2 \log (d+e x)+c e x \left (12 a e^2 (e x-2 d)+c \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )}{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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+c x^2\right )^2}{d+e x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 105, normalized size = 1.12 \begin {gather*} \frac {3 \, c^{2} e^{4} x^{4} - 4 \, c^{2} d e^{3} x^{3} + 6 \, {\left (c^{2} d^{2} e^{2} + 2 \, a c e^{4}\right )} x^{2} - 12 \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x + 12 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (e x + d\right )}{12 \, 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, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, 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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maple [A]  time = 0.04, size = 114, normalized size = 1.21 \begin {gather*} \frac {c^{2} x^{4}}{4 e}-\frac {c^{2} d \,x^{3}}{3 e^{2}}+\frac {a c \,x^{2}}{e}+\frac {c^{2} d^{2} x^{2}}{2 e^{3}}+\frac {a^{2} \ln \left (e x +d \right )}{e}+\frac {2 a c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {2 a c d x}{e^{2}}+\frac {c^{2} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {c^{2} d^{3} x}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.37, size = 105, normalized size = 1.12 \begin {gather*} \frac {3 \, c^{2} e^{3} x^{4} - 4 \, c^{2} d e^{2} x^{3} + 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}{12 \, e^{4}} + \frac {{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (e x + d\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, algorithm="maxima")

[Out]

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

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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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sympy [A]  time = 0.28, 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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