3.171 \(\int \frac{x^2 (c+d x)^{1+n}}{a+b x^3} \, dx\)

Optimal. Leaf size=253 \[ -\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right )}-\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c\right )}-\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right )} \]

[Out]

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

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Rubi [A]  time = 0.949081, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right )}-\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c\right )}-\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(c + d*x)^(1 + n))/(a + b*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 75.8272, size = 209, normalized size = 0.83 \[ - \frac{\left (c + d x\right )^{n + 2}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 2 \\ n + 3 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (c + d x\right )}{- \left (-1\right )^{\frac{2}{3}} \sqrt [3]{a} d + \sqrt [3]{b} c}} \right )}}{3 b^{\frac{2}{3}} \left (n + 2\right ) \left (- \left (-1\right )^{\frac{2}{3}} \sqrt [3]{a} d + \sqrt [3]{b} c\right )} - \frac{\left (c + d x\right )^{n + 2}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 2 \\ n + 3 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (c + d x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d + \sqrt [3]{b} c}} \right )}}{3 b^{\frac{2}{3}} \left (n + 2\right ) \left (\sqrt [3]{-1} \sqrt [3]{a} d + \sqrt [3]{b} c\right )} + \frac{\left (c + d x\right )^{n + 2}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 2 \\ n + 3 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (c + d x\right )}{- \sqrt [3]{a} d + \sqrt [3]{b} c}} \right )}}{3 b^{\frac{2}{3}} \left (n + 2\right ) \left (\sqrt [3]{a} d - \sqrt [3]{b} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(d*x+c)**(1+n)/(b*x**3+a),x)

[Out]

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

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Mathematica [C]  time = 0.327526, size = 375, normalized size = 1.48 \[ \frac{(c+d x)^n \left ((n+1) \left (b c^3-a d^3\right ) \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b c-3 \text{$\#$1} b c^2-a d^3+b c^3\&,\frac{\left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} c+c^2}\&\right ]+b \left (-2 c^2 (n+1) \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b c-3 \text{$\#$1} b c^2-a d^3+b c^3\&,\frac{\text{$\#$1} \left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} c+c^2}\&\right ]+c (n+1) \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b c-3 \text{$\#$1} b c^2-a d^3+b c^3\&,\frac{\text{$\#$1}^2 \left (\frac{c+d x}{-\text{$\#$1}+c+d x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{c+d x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} c+c^2}\&\right ]+3 n (c+d x)\right )\right )}{3 b^2 n (n+1)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^2*(c + d*x)^(1 + n))/(a + b*x^3),x]

[Out]

((c + d*x)^n*((b*c^3 - a*d^3)*(1 + n)*RootSum[b*c^3 - a*d^3 - 3*b*c^2*#1 + 3*b*c
*#1^2 - b*#1^3 & , Hypergeometric2F1[-n, -n, 1 - n, -(#1/(c + d*x - #1))]/(((c +
d*x)/(c + d*x - #1))^n*(c^2 - 2*c*#1 + #1^2)) & ] + b*(3*n*(c + d*x) - 2*c^2*(1
+ n)*RootSum[b*c^3 - a*d^3 - 3*b*c^2*#1 + 3*b*c*#1^2 - b*#1^3 & , (Hypergeometr
ic2F1[-n, -n, 1 - n, -(#1/(c + d*x - #1))]*#1)/(((c + d*x)/(c + d*x - #1))^n*(c^
2 - 2*c*#1 + #1^2)) & ] + c*(1 + n)*RootSum[b*c^3 - a*d^3 - 3*b*c^2*#1 + 3*b*c*#
1^2 - b*#1^3 & , (Hypergeometric2F1[-n, -n, 1 - n, -(#1/(c + d*x - #1))]*#1^2)/(
((c + d*x)/(c + d*x - #1))^n*(c^2 - 2*c*#1 + #1^2)) & ])))/(3*b^2*n*(1 + n))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( dx+c \right ) ^{1+n}}{b{x}^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(d*x+c)^(1+n)/(b*x^3+a),x)

[Out]

int(x^2*(d*x+c)^(1+n)/(b*x^3+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(n + 1)*x^2/(b*x^3 + a),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(n + 1)*x^2/(b*x^3 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(n + 1)*x^2/(b*x^3 + a),x, algorithm="fricas")

[Out]

integral((d*x + c)^(n + 1)*x^2/(b*x^3 + 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**2*(d*x+c)**(1+n)/(b*x**3+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(n + 1)*x^2/(b*x^3 + a),x, algorithm="giac")

[Out]

integrate((d*x + c)^(n + 1)*x^2/(b*x^3 + a), x)