3.5.4 \(\int \frac {x (d+e x^2)^q}{a+b x^2+c x^4} \, dx\) [404]

Optimal. Leaf size=198 \[ -\frac {c \left (d+e x^2\right )^{1+q} \, _2F_1\left (1,1+q;2+q;\frac {2 c \left (d+e x^2\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1+q)}+\frac {c \left (d+e x^2\right )^{1+q} \, _2F_1\left (1,1+q;2+q;\frac {2 c \left (d+e x^2\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+q)} \]

[Out]

-c*(e*x^2+d)^(1+q)*hypergeom([1, 1+q],[2+q],2*c*(e*x^2+d)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2))))/(1+q)/(2*c*d-e*(b-
(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(1/2)+c*(e*x^2+d)^(1+q)*hypergeom([1, 1+q],[2+q],2*c*(e*x^2+d)/(2*c*d-e*(b+(
-4*a*c+b^2)^(1/2))))/(1+q)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))

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Rubi [A]
time = 0.26, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1261, 725, 70} \begin {gather*} \frac {c \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac {2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{(q+1) \sqrt {b^2-4 a c} \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {c \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac {2 c \left (e x^2+d\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{(q+1) \sqrt {b^2-4 a c} \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]

[Out]

-((c*(d + e*x^2)^(1 + q)*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])
*e)])/(Sqrt[b^2 - 4*a*c]*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(1 + q))) + (c*(d + e*x^2)^(1 + q)*Hypergeometric
2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[b^2 - 4*a*c]*(2*c*d - (b +
Sqrt[b^2 - 4*a*c])*e)*(1 + q))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 725

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^
m, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a
*e^2, 0] && NeQ[2*c*d - b*e, 0] &&  !IntegerQ[m]

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rubi steps

\begin {align*} \int \frac {x \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(d+e x)^q}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {2 c (d+e x)^q}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}-\frac {2 c (d+e x)^q}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {c \text {Subst}\left (\int \frac {(d+e x)^q}{b-\sqrt {b^2-4 a c}+2 c x} \, dx,x,x^2\right )}{\sqrt {b^2-4 a c}}-\frac {c \text {Subst}\left (\int \frac {(d+e x)^q}{b+\sqrt {b^2-4 a c}+2 c x} \, dx,x,x^2\right )}{\sqrt {b^2-4 a c}}\\ &=-\frac {c \left (d+e x^2\right )^{1+q} \, _2F_1\left (1,1+q;2+q;\frac {2 c \left (d+e x^2\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1+q)}+\frac {c \left (d+e x^2\right )^{1+q} \, _2F_1\left (1,1+q;2+q;\frac {2 c \left (d+e x^2\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+q)}\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 168, normalized size = 0.85 \begin {gather*} \frac {c \left (d+e x^2\right )^{1+q} \left (-\frac {\, _2F_1\left (1,1+q;2+q;\frac {2 c \left (d+e x^2\right )}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}+\frac {\, _2F_1\left (1,1+q;2+q;\frac {2 c \left (d+e x^2\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} (1+q)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(d + e*x^2)^(1 + q)*(-(Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c
])*e)]/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)) + Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - (
b + Sqrt[b^2 - 4*a*c])*e)]/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)))/(Sqrt[b^2 - 4*a*c]*(1 + q))

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x \left (e \,x^{2}+d \right )^{q}}{c \,x^{4}+b \,x^{2}+a}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x^2+d)^q/(c*x^4+b*x^2+a),x)

[Out]

int(x*(e*x^2+d)^q/(c*x^4+b*x^2+a),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^2*e + d)^q*x/(c*x^4 + b*x^2 + a), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((x^2*e + d)^q*x/(c*x^4 + b*x^2 + a), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^2*e + d)^q*x/(c*x^4 + b*x^2 + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,{\left (e\,x^2+d\right )}^q}{c\,x^4+b\,x^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x)

[Out]

int((x*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x)

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