3.375 \(\int \frac{(d+e x)^n}{x \left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=489 \[ -\frac{\sqrt{c} e n \left (\sqrt{-a} \sqrt{c} d+a e\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 a^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 a^2 (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 a^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{e x}{d}+1\right )}{a^2 d (n+1)}+\frac{\sqrt{c} e n \left (\sqrt{-a} e+\sqrt{c} d\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 (-a)^{3/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{c (d-e x) (d+e x)^{n+1}}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]

[Out]

(c*(d - e*x)*(d + e*x)^(1 + n))/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) + (Sqrt[c]*(d
+ e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d
- Sqrt[-a]*e)])/(2*a^2*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) + (Sqrt[c]*e*(Sqrt[c]*
d + Sqrt[-a]*e)*n*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*
(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(4*(-a)^(3/2)*(Sqrt[c]*d - Sqrt[-a]*e)*(c*
d^2 + a*e^2)*(1 + n)) + (Sqrt[c]*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2
+ n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*a^2*(Sqrt[c]*d + Sqrt[-a
]*e)*(1 + n)) - (Sqrt[c]*e*(Sqrt[-a]*Sqrt[c]*d + a*e)*n*(d + e*x)^(1 + n)*Hyperg
eometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(4*a
^2*(Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + n)) - ((d + e*x)^(1 + n)*Hyperg
eometric2F1[1, 1 + n, 2 + n, 1 + (e*x)/d])/(a^2*d*(1 + n))

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Rubi [A]  time = 1.24952, antiderivative size = 489, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{c} e n \left (\sqrt{-a} \sqrt{c} d+a e\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 a^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 a^2 (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 a^2 (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{e x}{d}+1\right )}{a^2 d (n+1)}+\frac{\sqrt{c} e n \left (\sqrt{-a} e+\sqrt{c} d\right ) (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 (-a)^{3/2} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{c (d-e x) (d+e x)^{n+1}}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d - e*x)*(d + e*x)^(1 + n))/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) + (Sqrt[c]*(d
+ e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d
- Sqrt[-a]*e)])/(2*a^2*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) + (Sqrt[c]*e*(Sqrt[c]*
d + Sqrt[-a]*e)*n*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*
(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(4*(-a)^(3/2)*(Sqrt[c]*d - Sqrt[-a]*e)*(c*
d^2 + a*e^2)*(1 + n)) + (Sqrt[c]*(d + e*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2
+ n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*a^2*(Sqrt[c]*d + Sqrt[-a
]*e)*(1 + n)) - (Sqrt[c]*e*(Sqrt[-a]*Sqrt[c]*d + a*e)*n*(d + e*x)^(1 + n)*Hyperg
eometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(4*a
^2*(Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + n)) - ((d + e*x)^(1 + n)*Hyperg
eometric2F1[1, 1 + n, 2 + n, 1 + (e*x)/d])/(a^2*d*(1 + n))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.0977494, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^n}{x \left (a+c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{n}}{x \left ( c{x}^{2}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^n/x/(c*x^2+a)^2,x)

[Out]

int((e*x+d)^n/x/(c*x^2+a)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^n/((c*x^2 + a)^2*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{n}}{c^{2} x^{5} + 2 \, a c x^{3} + a^{2} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)^n/(c^2*x^5 + 2*a*c*x^3 + a^2*x), 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((e*x+d)**n/x/(c*x**2+a)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^n/((c*x^2 + a)^2*x), x)