3.203 \(\int \frac{x^3 (c+d x)^{1+n}}{a+b x^4} \, dx\)

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

[Out]

-((c + d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + 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)*(2 + n))
- ((c + d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + 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)*(2 + n
)) - ((c + d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + 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)*(2 + n)) - ((c
+ d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + 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)*(2 + n))

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Rubi [A]  time = 1.2044, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/4} (n+2) \left (\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d\right )}-\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{4 b^{3/4} (n+2) \left (\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c\right )}-\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+2) \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right )}-\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+2) \left (\sqrt [4]{-a} d+\sqrt [4]{b} c\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(c + d*x)^(1 + n))/(a + b*x^4),x]

[Out]

-((c + d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + 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)*(2 + n))
- ((c + d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + 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)*(2 + n
)) - ((c + d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + 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)*(2 + n)) - ((c
+ d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + 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)*(2 + n))

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Rubi in Sympy [A]  time = 90.0766, size = 265, normalized size = 0.76 \[ - \frac{\left (c + d x\right )^{n + 2}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 2 \\ n + 3 \end{matrix}\middle |{\frac{\sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{b} c + i d \sqrt [4]{- a}}} \right )}}{4 b^{\frac{3}{4}} \left (n + 2\right ) \left (\sqrt [4]{b} c + i d \sqrt [4]{- a}\right )} - \frac{\left (c + d x\right )^{n + 2}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 2 \\ n + 3 \end{matrix}\middle |{\frac{\sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{b} c - i d \sqrt [4]{- a}}} \right )}}{4 b^{\frac{3}{4}} \left (n + 2\right ) \left (\sqrt [4]{b} c - i d \sqrt [4]{- a}\right )} - \frac{\left (c + d x\right )^{n + 2}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 2 \\ n + 3 \end{matrix}\middle |{\frac{\sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{b} c + d \sqrt [4]{- a}}} \right )}}{4 b^{\frac{3}{4}} \left (n + 2\right ) \left (\sqrt [4]{b} c + d \sqrt [4]{- a}\right )} - \frac{\left (c + d x\right )^{n + 2}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 2 \\ n + 3 \end{matrix}\middle |{\frac{\sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{b} c - d \sqrt [4]{- a}}} \right )}}{4 b^{\frac{3}{4}} \left (n + 2\right ) \left (\sqrt [4]{b} c - d \sqrt [4]{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(d*x+c)**(1+n)/(b*x**4+a),x)

[Out]

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

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Mathematica [C]  time = 0.348139, size = 691, normalized size = 1.98 \[ \frac{(c+d x)^n \left ((n+1) \left (a d^4+b c^4\right ) \text{RootSum}\left [\text{$\#$1}^4 b-4 \text{$\#$1}^3 b c+6 \text{$\#$1}^2 b c^2-4 \text{$\#$1} b c^3+a d^4+b c^4\&,\frac{\left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{-\text{$\#$1}^3+3 \text{$\#$1}^2 c-3 \text{$\#$1} c^2+c^3}\&\right ]-b \left (3 c^3 (n+1) \text{RootSum}\left [\text{$\#$1}^4 b-4 \text{$\#$1}^3 b c+6 \text{$\#$1}^2 b c^2-4 \text{$\#$1} b c^3+a d^4+b c^4\&,\frac{\text{$\#$1} \left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{-\text{$\#$1}^3+3 \text{$\#$1}^2 c-3 \text{$\#$1} c^2+c^3}\&\right ]-3 c^2 (n+1) \text{RootSum}\left [\text{$\#$1}^4 b-4 \text{$\#$1}^3 b c+6 \text{$\#$1}^2 b c^2-4 \text{$\#$1} b c^3+a d^4+b c^4\&,\frac{\text{$\#$1}^2 \left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{-\text{$\#$1}^3+3 \text{$\#$1}^2 c-3 \text{$\#$1} c^2+c^3}\&\right ]+c n \text{RootSum}\left [\text{$\#$1}^4 b-4 \text{$\#$1}^3 b c+6 \text{$\#$1}^2 b c^2-4 \text{$\#$1} b c^3+a d^4+b c^4\&,\frac{\text{$\#$1}^3 \left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{-\text{$\#$1}^3+3 \text{$\#$1}^2 c-3 \text{$\#$1} c^2+c^3}\&\right ]+c \text{RootSum}\left [\text{$\#$1}^4 b-4 \text{$\#$1}^3 b c+6 \text{$\#$1}^2 b c^2-4 \text{$\#$1} b c^3+a d^4+b c^4\&,\frac{\text{$\#$1}^3 \left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{-\text{$\#$1}^3+3 \text{$\#$1}^2 c-3 \text{$\#$1} c^2+c^3}\&\right ]-4 c n-4 d n x\right )\right )}{4 b^2 n (n+1)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^3*(c + d*x)^(1 + n))/(a + b*x^4),x]

[Out]

((c + d*x)^n*((b*c^4 + a*d^4)*(1 + n)*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)) & ] -
b*(-4*c*n - 4*d*n*x + 3*c^3*(1 + n)*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)/(((c + d*x)/(c + d*x - #1))^n*(c^3 - 3*c^2*#1 + 3*c*#1^2 - #1^3)) &
] - 3*c^2*(1 + n)*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)) & ] + 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 & , (Hypergeometr
ic2F1[-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)) & ] + c*n*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^2*n*(1 + n))

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( dx+c \right ) ^{1+n}}{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(d*x+c)^(1+n)/(b*x^4+a),x)

[Out]

int(x^3*(d*x+c)^(1+n)/(b*x^4+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{n + 1} x^{3}}{b x^{4} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(n + 1)*x^3/(b*x^4 + a),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(n + 1)*x^3/(b*x^4 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{n + 1} x^{3}}{b x^{4} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(n + 1)*x^3/(b*x^4 + a),x, algorithm="fricas")

[Out]

integral((d*x + c)^(n + 1)*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)**(1+n)/(b*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{n + 1} x^{3}}{b x^{4} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(n + 1)*x^3/(b*x^4 + a),x, algorithm="giac")

[Out]

integrate((d*x + c)^(n + 1)*x^3/(b*x^4 + a), x)