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

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 = 1.46328, 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 \[ -\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d\right )}-\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c\right )}-\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right )}-\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{-a} d+\sqrt [4]{b} c\right )} \]

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))

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Rubi in Sympy [A]  time = 86.3924, size = 265, normalized size = 0.76 \[ - \frac{\left (c + d x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{b} c + i d \sqrt [4]{- a}}} \right )}}{4 b^{\frac{3}{4}} \left (n + 1\right ) \left (\sqrt [4]{b} c + i d \sqrt [4]{- a}\right )} - \frac{\left (c + d x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{b} c - i d \sqrt [4]{- a}}} \right )}}{4 b^{\frac{3}{4}} \left (n + 1\right ) \left (\sqrt [4]{b} c - i d \sqrt [4]{- a}\right )} - \frac{\left (c + d x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{b} c + d \sqrt [4]{- a}}} \right )}}{4 b^{\frac{3}{4}} \left (n + 1\right ) \left (\sqrt [4]{b} c + d \sqrt [4]{- a}\right )} - \frac{\left (c + d x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{b} c - d \sqrt [4]{- a}}} \right )}}{4 b^{\frac{3}{4}} \left (n + 1\right ) \left (\sqrt [4]{b} c - d \sqrt [4]{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c + d*x)**(n + 1)*hyper((1, n + 1), (n + 2,), b**(1/4)*(c + d*x)/(b**(1/4)*c +
I*d*(-a)**(1/4)))/(4*b**(3/4)*(n + 1)*(b**(1/4)*c + I*d*(-a)**(1/4))) - (c + d*
x)**(n + 1)*hyper((1, n + 1), (n + 2,), b**(1/4)*(c + d*x)/(b**(1/4)*c - I*d*(-a
)**(1/4)))/(4*b**(3/4)*(n + 1)*(b**(1/4)*c - I*d*(-a)**(1/4))) - (c + d*x)**(n +
1)*hyper((1, n + 1), (n + 2,), b**(1/4)*(c + d*x)/(b**(1/4)*c + d*(-a)**(1/4)))
/(4*b**(3/4)*(n + 1)*(b**(1/4)*c + d*(-a)**(1/4))) - (c + d*x)**(n + 1)*hyper((1
, n + 1), (n + 2,), b**(1/4)*(c + d*x)/(b**(1/4)*c - d*(-a)**(1/4)))/(4*b**(3/4)
*(n + 1)*(b**(1/4)*c - d*(-a)**(1/4)))

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Mathematica [C]  time = 0.253223, size = 526, normalized size = 1.51 \[ \frac{(c+d x)^n \left (c^3 \text{RootSum}\left [\text{$\#$1}^4 b-4 \text{$\#$1}^3 b c+6 \text{$\#$1}^2 b c^2-4 \text{$\#$1} b c^3+a d^4+b c^4\&,\frac{\left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{-\text{$\#$1}^3+3 \text{$\#$1}^2 c-3 \text{$\#$1} c^2+c^3}\&\right ]-3 c^2 \text{RootSum}\left [\text{$\#$1}^4 b-4 \text{$\#$1}^3 b c+6 \text{$\#$1}^2 b c^2-4 \text{$\#$1} b c^3+a d^4+b c^4\&,\frac{\text{$\#$1} \left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{-\text{$\#$1}^3+3 \text{$\#$1}^2 c-3 \text{$\#$1} c^2+c^3}\&\right ]+3 c \text{RootSum}\left [\text{$\#$1}^4 b-4 \text{$\#$1}^3 b c+6 \text{$\#$1}^2 b c^2-4 \text{$\#$1} b c^3+a d^4+b c^4\&,\frac{\text{$\#$1}^2 \left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{-\text{$\#$1}^3+3 \text{$\#$1}^2 c-3 \text{$\#$1} c^2+c^3}\&\right ]-\text{RootSum}\left [\text{$\#$1}^4 b-4 \text{$\#$1}^3 b c+6 \text{$\#$1}^2 b c^2-4 \text{$\#$1} b c^3+a d^4+b c^4\&,\frac{\text{$\#$1}^3 \left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{-\text{$\#$1}^3+3 \text{$\#$1}^2 c-3 \text{$\#$1} c^2+c^3}\&\right ]\right )}{4 b n} \]

Warning: Unable to verify antiderivative.

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

[Out]

((c + d*x)^n*(c^3*RootSum[b*c^4 + a*d^4 - 4*b*c^3*#1 + 6*b*c^2*#1^2 - 4*b*c*#1^3
+ b*#1^4 & , Hypergeometric2F1[-n, -n, 1 - n, -(#1/(c + d*x - #1))]/(((c + d*x)
/(c + d*x - #1))^n*(c^3 - 3*c^2*#1 + 3*c*#1^2 - #1^3)) & ] - 3*c^2*RootSum[b*c^4
+ a*d^4 - 4*b*c^3*#1 + 6*b*c^2*#1^2 - 4*b*c*#1^3 + b*#1^4 & , (Hypergeometric2F
1[-n, -n, 1 - n, -(#1/(c + d*x - #1))]*#1)/(((c + d*x)/(c + d*x - #1))^n*(c^3 -
3*c^2*#1 + 3*c*#1^2 - #1^3)) & ] + 3*c*RootSum[b*c^4 + a*d^4 - 4*b*c^3*#1 + 6*b*
c^2*#1^2 - 4*b*c*#1^3 + b*#1^4 & , (Hypergeometric2F1[-n, -n, 1 - n, -(#1/(c + d
*x - #1))]*#1^2)/(((c + d*x)/(c + d*x - #1))^n*(c^3 - 3*c^2*#1 + 3*c*#1^2 - #1^3
)) & ] - RootSum[b*c^4 + a*d^4 - 4*b*c^3*#1 + 6*b*c^2*#1^2 - 4*b*c*#1^3 + b*#1^4
& , (Hypergeometric2F1[-n, -n, 1 - n, -(#1/(c + d*x - #1))]*#1^3)/(((c + d*x)/(
c + d*x - #1))^n*(c^3 - 3*c^2*#1 + 3*c*#1^2 - #1^3)) & ]))/(4*b*n)

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( dx+c \right ) ^{n}}{b{x}^{4}+a}}\, dx \]

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. \[ \int \frac{{\left (d x + c\right )}^{n} x^{3}}{b x^{4} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n*x^3/(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. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{n} x^{3}}{b x^{4} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n*x^3/(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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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