3.224 \(\int \frac{x^3 (c+d x)^n}{a+b x^4} \, dx\)

Optimal. Leaf size=349 \[ -\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d\right )}-\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c\right )}-\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right )}-\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{-a} d+\sqrt [4]{b} c\right )} \]

[Out]

-((c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(
4*b^(3/4)*(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)*(1 + n)) - ((c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^
(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*b^(3/4)*(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)*(1 + n)) - ((c +
d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*b^(3/4)*(
b^(1/4)*c - (-a)^(1/4)*d)*(1 + n)) - ((c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))
/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*b^(3/4)*(b^(1/4)*c + (-a)^(1/4)*d)*(1 + n))

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Rubi [A]  time = 0.729258, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6725, 831, 68} \[ -\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d\right )}-\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c\right )}-\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right )}-\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{-a} d+\sqrt [4]{b} c\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x)^n)/(a + b*x^4),x]

[Out]

-((c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(
4*b^(3/4)*(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)*(1 + n)) - ((c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^
(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*b^(3/4)*(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)*(1 + n)) - ((c +
d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*b^(3/4)*(
b^(1/4)*c - (-a)^(1/4)*d)*(1 + n)) - ((c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))
/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*b^(3/4)*(b^(1/4)*c + (-a)^(1/4)*d)*(1 + n))

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 831

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] &&  !Ration
alQ[m]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{x^3 (c+d x)^n}{a+b x^4} \, dx &=\int \left (\frac{x (c+d x)^n}{2 \left (-\sqrt{-a} \sqrt{b}+b x^2\right )}+\frac{x (c+d x)^n}{2 \left (\sqrt{-a} \sqrt{b}+b x^2\right )}\right ) \, dx\\ &=\frac{1}{2} \int \frac{x (c+d x)^n}{-\sqrt{-a} \sqrt{b}+b x^2} \, dx+\frac{1}{2} \int \frac{x (c+d x)^n}{\sqrt{-a} \sqrt{b}+b x^2} \, dx\\ &=\frac{1}{2} \int \left (-\frac{(c+d x)^n}{2 b^{3/4} \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}+\frac{(c+d x)^n}{2 b^{3/4} \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}\right ) \, dx+\frac{1}{2} \int \left (-\frac{(c+d x)^n}{2 b^{3/4} \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}+\frac{(c+d x)^n}{2 b^{3/4} \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{(c+d x)^n}{\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x} \, dx}{4 b^{3/4}}-\frac{\int \frac{(c+d x)^n}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 b^{3/4}}+\frac{\int \frac{(c+d x)^n}{\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x} \, dx}{4 b^{3/4}}+\frac{\int \frac{(c+d x)^n}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 b^{3/4}}\\ &=-\frac{(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d\right ) (1+n)}-\frac{(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d\right ) (1+n)}-\frac{(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right ) (1+n)}-\frac{(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt [4]{-a} d\right ) (1+n)}\\ \end{align*}

Mathematica [C]  time = 0.393453, size = 274, normalized size = 0.79 \[ \frac{(c+d x)^{n+1} \left (-\frac{\, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}-\frac{\, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}-\frac{\, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}-\frac{\, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^{3/4} (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c + d*x)^(1 + n)*(-(Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(b^(1
/4)*c - (-a)^(1/4)*d)) - Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)]/
(b^(1/4)*c - I*(-a)^(1/4)*d) - Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a)^(1/4
)*d)]/(b^(1/4)*c + I*(-a)^(1/4)*d) - Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^
(1/4)*d)]/(b^(1/4)*c + (-a)^(1/4)*d)))/(4*b^(3/4)*(1 + n))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3} \left ( dx+c \right ) ^{n}}{b{x}^{4}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(d*x+c)^n/(b*x^4+a),x)

[Out]

int(x^3*(d*x+c)^n/(b*x^4+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{n} x^{3}}{b x^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^n*x^3/(b*x^4 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d x + c\right )}^{n} x^{3}}{b x^{4} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*x + c)^n*x^3/(b*x^4 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(d*x+c)**n/(b*x**4+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{n} x^{3}}{b x^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^n*x^3/(b*x^4 + a), x)