3.1894 \(\int (d+e x)^{-3-2 p} (f+g x) (a^2+2 a b x+b^2 x^2)^p \, dx\)

Optimal. Leaf size=119 \[ \frac {(a+b x) (b f-a g) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 p-1}}{(2 p+1) (b d-a e)^2}-\frac {\left (a^2+2 a b x+b^2 x^2\right )^{p+1} (e f-d g) (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)^2} \]

[Out]

(-a*g+b*f)*(b*x+a)*(e*x+d)^(-1-2*p)*(b^2*x^2+2*a*b*x+a^2)^p/(-a*e+b*d)^2/(1+2*p)-1/2*(-d*g+e*f)*(b^2*x^2+2*a*b
*x+a^2)^(1+p)/(-a*e+b*d)^2/(1+p)/((e*x+d)^(2+2*p))

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Rubi [A]  time = 0.07, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {769, 646, 37} \[ \frac {(a+b x) (b f-a g) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 p-1}}{(2 p+1) (b d-a e)^2}-\frac {\left (a^2+2 a b x+b^2 x^2\right )^{p+1} (e f-d g) (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(-3 - 2*p)*(f + g*x)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

((b*f - a*g)*(a + b*x)*(d + e*x)^(-1 - 2*p)*(a^2 + 2*a*b*x + b^2*x^2)^p)/((b*d - a*e)^2*(1 + 2*p)) - ((e*f - d
*g)*(a^2 + 2*a*b*x + b^2*x^2)^(1 + p))/(2*(b*d - a*e)^2*(1 + p)*(d + e*x)^(2*(1 + p)))

Rule 37

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

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rule 769

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(-2*c*(e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)^2), x] + Dist[(2*c*f -
b*g)/(2*c*d - b*e), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x]
&& EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && NeQ[2*c*f - b*g, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int (d+e x)^{-3-2 p} (f+g x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=-\frac {(e f-d g) (d+e x)^{-2 (1+p)} \left (a^2+2 a b x+b^2 x^2\right )^{1+p}}{2 (b d-a e)^2 (1+p)}+\frac {(b f-a g) \int (d+e x)^{-2-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx}{b d-a e}\\ &=-\frac {(e f-d g) (d+e x)^{-2 (1+p)} \left (a^2+2 a b x+b^2 x^2\right )^{1+p}}{2 (b d-a e)^2 (1+p)}+\frac {\left ((b f-a g) \left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (a b+b^2 x\right )^{2 p} (d+e x)^{-2-2 p} \, dx}{b d-a e}\\ &=\frac {(b f-a g) (a+b x) (d+e x)^{-1-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p}{(b d-a e)^2 (1+2 p)}-\frac {(e f-d g) (d+e x)^{-2 (1+p)} \left (a^2+2 a b x+b^2 x^2\right )^{1+p}}{2 (b d-a e)^2 (1+p)}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 97, normalized size = 0.82 \[ \frac {(a+b x) \left ((a+b x)^2\right )^p (d+e x)^{-2 (p+1)} (b (2 d f (p+1)+d g (2 p+1) x+e f x)-a (d g+e (2 f p+f+2 g (p+1) x)))}{2 (p+1) (2 p+1) (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(-3 - 2*p)*(f + g*x)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

((a + b*x)*((a + b*x)^2)^p*(b*(2*d*f*(1 + p) + e*f*x + d*g*(1 + 2*p)*x) - a*(d*g + e*(f + 2*f*p + 2*g*(1 + p)*
x))))/(2*(b*d - a*e)^2*(1 + p)*(1 + 2*p)*(d + e*x)^(2*(1 + p)))

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fricas [B]  time = 1.79, size = 348, normalized size = 2.92 \[ -\frac {{\left (a^{2} d^{2} g - {\left (b^{2} e^{2} f + 2 \, {\left (b^{2} d e - a b e^{2}\right )} g p + {\left (b^{2} d e - 2 \, a b e^{2}\right )} g\right )} x^{3} - 2 \, {\left (a b d^{2} - a^{2} d e\right )} f p - {\left (3 \, b^{2} d e f + {\left (b^{2} d^{2} - 2 \, a b d e - 2 \, a^{2} e^{2}\right )} g + 2 \, {\left ({\left (b^{2} d e - a b e^{2}\right )} f + {\left (b^{2} d^{2} - a^{2} e^{2}\right )} g\right )} p\right )} x^{2} - {\left (2 \, a b d^{2} - a^{2} d e\right )} f + {\left (3 \, a^{2} d e g - {\left (2 \, b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2}\right )} f - 2 \, {\left ({\left (b^{2} d^{2} - a^{2} e^{2}\right )} f + {\left (a b d^{2} - a^{2} d e\right )} g\right )} p\right )} x\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}}{2 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2} + 2 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} p^{2} + 3 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} p\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(g*x+f)*(b^2*x^2+2*a*b*x+a^2)^p,x, algorithm="fricas")

[Out]

-1/2*(a^2*d^2*g - (b^2*e^2*f + 2*(b^2*d*e - a*b*e^2)*g*p + (b^2*d*e - 2*a*b*e^2)*g)*x^3 - 2*(a*b*d^2 - a^2*d*e
)*f*p - (3*b^2*d*e*f + (b^2*d^2 - 2*a*b*d*e - 2*a^2*e^2)*g + 2*((b^2*d*e - a*b*e^2)*f + (b^2*d^2 - a^2*e^2)*g)
*p)*x^2 - (2*a*b*d^2 - a^2*d*e)*f + (3*a^2*d*e*g - (2*b^2*d^2 + 2*a*b*d*e - a^2*e^2)*f - 2*((b^2*d^2 - a^2*e^2
)*f + (a*b*d^2 - a^2*d*e)*g)*p)*x)*(b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^(-2*p - 3)/(b^2*d^2 - 2*a*b*d*e + a^2
*e^2 + 2*(b^2*d^2 - 2*a*b*d*e + a^2*e^2)*p^2 + 3*(b^2*d^2 - 2*a*b*d*e + a^2*e^2)*p)

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giac [B]  time = 0.34, size = 1352, normalized size = 11.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(g*x+f)*(b^2*x^2+2*a*b*x+a^2)^p,x, algorithm="giac")

[Out]

1/2*(2*(b^2*x^2 + 2*a*b*x + a^2)^p*b^2*d*g*p*x^3*e^(-2*p*log(x*e + d) - 3*log(x*e + d) + 1) + 2*(b^2*x^2 + 2*a
*b*x + a^2)^p*b^2*d^2*g*p*x^2*e^(-2*p*log(x*e + d) - 3*log(x*e + d)) - 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b*g*p*x
^3*e^(-2*p*log(x*e + d) - 3*log(x*e + d) + 2) + 2*(b^2*x^2 + 2*a*b*x + a^2)^p*b^2*d*f*p*x^2*e^(-2*p*log(x*e +
d) - 3*log(x*e + d) + 1) + (b^2*x^2 + 2*a*b*x + a^2)^p*b^2*d*g*x^3*e^(-2*p*log(x*e + d) - 3*log(x*e + d) + 1)
+ 2*(b^2*x^2 + 2*a*b*x + a^2)^p*b^2*d^2*f*p*x*e^(-2*p*log(x*e + d) - 3*log(x*e + d)) + 2*(b^2*x^2 + 2*a*b*x +
a^2)^p*a*b*d^2*g*p*x*e^(-2*p*log(x*e + d) - 3*log(x*e + d)) + (b^2*x^2 + 2*a*b*x + a^2)^p*b^2*d^2*g*x^2*e^(-2*
p*log(x*e + d) - 3*log(x*e + d)) - 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b*f*p*x^2*e^(-2*p*log(x*e + d) - 3*log(x*e
+ d) + 2) - 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*g*p*x^2*e^(-2*p*log(x*e + d) - 3*log(x*e + d) + 2) + (b^2*x^2 +
2*a*b*x + a^2)^p*b^2*f*x^3*e^(-2*p*log(x*e + d) - 3*log(x*e + d) + 2) - 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b*g*x^
3*e^(-2*p*log(x*e + d) - 3*log(x*e + d) + 2) - 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*d*g*p*x*e^(-2*p*log(x*e + d)
- 3*log(x*e + d) + 1) + 3*(b^2*x^2 + 2*a*b*x + a^2)^p*b^2*d*f*x^2*e^(-2*p*log(x*e + d) - 3*log(x*e + d) + 1) -
2*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b*d*g*x^2*e^(-2*p*log(x*e + d) - 3*log(x*e + d) + 1) + 2*(b^2*x^2 + 2*a*b*x +
a^2)^p*a*b*d^2*f*p*e^(-2*p*log(x*e + d) - 3*log(x*e + d)) + 2*(b^2*x^2 + 2*a*b*x + a^2)^p*b^2*d^2*f*x*e^(-2*p
*log(x*e + d) - 3*log(x*e + d)) - 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*f*p*x*e^(-2*p*log(x*e + d) - 3*log(x*e + d
) + 2) - 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*g*x^2*e^(-2*p*log(x*e + d) - 3*log(x*e + d) + 2) - 2*(b^2*x^2 + 2*a
*b*x + a^2)^p*a^2*d*f*p*e^(-2*p*log(x*e + d) - 3*log(x*e + d) + 1) + 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b*d*f*x*e
^(-2*p*log(x*e + d) - 3*log(x*e + d) + 1) - 3*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*d*g*x*e^(-2*p*log(x*e + d) - 3*l
og(x*e + d) + 1) + 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b*d^2*f*e^(-2*p*log(x*e + d) - 3*log(x*e + d)) - (b^2*x^2 +
2*a*b*x + a^2)^p*a^2*d^2*g*e^(-2*p*log(x*e + d) - 3*log(x*e + d)) - (b^2*x^2 + 2*a*b*x + a^2)^p*a^2*f*x*e^(-2
*p*log(x*e + d) - 3*log(x*e + d) + 2) - (b^2*x^2 + 2*a*b*x + a^2)^p*a^2*d*f*e^(-2*p*log(x*e + d) - 3*log(x*e +
d) + 1))/(2*b^2*d^2*p^2 - 4*a*b*d*p^2*e + 3*b^2*d^2*p + 2*a^2*p^2*e^2 - 6*a*b*d*p*e + b^2*d^2 + 3*a^2*p*e^2 -
2*a*b*d*e + a^2*e^2)

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maple [A]  time = 0.05, size = 174, normalized size = 1.46 \[ -\frac {\left (b x +a \right ) \left (2 a e g p x -2 b d g p x +2 a e f p +2 a e g x -2 b d f p -b d g x -b e f x +a d g +a e f -2 b d f \right ) \left (e x +d \right )^{-2 p -2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{2 \left (2 a^{2} e^{2} p^{2}-4 a b d e \,p^{2}+2 b^{2} d^{2} p^{2}+3 a^{2} e^{2} p -6 a b d e p +3 b^{2} d^{2} p +a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(-2*p-3)*(g*x+f)*(b^2*x^2+2*a*b*x+a^2)^p,x)

[Out]

-1/2*(b*x+a)*(e*x+d)^(-2*p-2)*(2*a*e*g*p*x-2*b*d*g*p*x+2*a*e*f*p+2*a*e*g*x-2*b*d*f*p-b*d*g*x-b*e*f*x+a*d*g+a*e
*f-2*b*d*f)*(b^2*x^2+2*a*b*x+a^2)^p/(2*a^2*e^2*p^2-4*a*b*d*e*p^2+2*b^2*d^2*p^2+3*a^2*e^2*p-6*a*b*d*e*p+3*b^2*d
^2*p+a^2*e^2-2*a*b*d*e+b^2*d^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (g x + f\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(g*x+f)*(b^2*x^2+2*a*b*x+a^2)^p,x, algorithm="maxima")

[Out]

integrate((g*x + f)*(b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^(-2*p - 3), x)

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mupad [B]  time = 2.43, size = 375, normalized size = 3.15 \[ -{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {x^2\,\left (2\,a^2\,e^2\,g-b^2\,d^2\,g-3\,b^2\,d\,e\,f+2\,a^2\,e^2\,g\,p-2\,b^2\,d^2\,g\,p+2\,a\,b\,d\,e\,g+2\,a\,b\,e^2\,f\,p-2\,b^2\,d\,e\,f\,p\right )}{2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (2\,p^2+3\,p+1\right )}+\frac {x\,\left (a^2\,e^2\,f-2\,b^2\,d^2\,f+3\,a^2\,d\,e\,g+2\,a^2\,e^2\,f\,p-2\,b^2\,d^2\,f\,p-2\,a\,b\,d\,e\,f-2\,a\,b\,d^2\,g\,p+2\,a^2\,d\,e\,g\,p\right )}{2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (2\,p^2+3\,p+1\right )}+\frac {a\,d\,\left (a\,d\,g+a\,e\,f-2\,b\,d\,f+2\,a\,e\,f\,p-2\,b\,d\,f\,p\right )}{2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (2\,p^2+3\,p+1\right )}-\frac {b\,e\,x^3\,\left (b\,d\,g-2\,a\,e\,g+b\,e\,f-2\,a\,e\,g\,p+2\,b\,d\,g\,p\right )}{2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (2\,p^2+3\,p+1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(a^2 + b^2*x^2 + 2*a*b*x)^p)/(d + e*x)^(2*p + 3),x)

[Out]

-(a^2 + b^2*x^2 + 2*a*b*x)^p*((x^2*(2*a^2*e^2*g - b^2*d^2*g - 3*b^2*d*e*f + 2*a^2*e^2*g*p - 2*b^2*d^2*g*p + 2*
a*b*d*e*g + 2*a*b*e^2*f*p - 2*b^2*d*e*f*p))/(2*(a*e - b*d)^2*(d + e*x)^(2*p + 3)*(3*p + 2*p^2 + 1)) + (x*(a^2*
e^2*f - 2*b^2*d^2*f + 3*a^2*d*e*g + 2*a^2*e^2*f*p - 2*b^2*d^2*f*p - 2*a*b*d*e*f - 2*a*b*d^2*g*p + 2*a^2*d*e*g*
p))/(2*(a*e - b*d)^2*(d + e*x)^(2*p + 3)*(3*p + 2*p^2 + 1)) + (a*d*(a*d*g + a*e*f - 2*b*d*f + 2*a*e*f*p - 2*b*
d*f*p))/(2*(a*e - b*d)^2*(d + e*x)^(2*p + 3)*(3*p + 2*p^2 + 1)) - (b*e*x^3*(b*d*g - 2*a*e*g + b*e*f - 2*a*e*g*
p + 2*b*d*g*p))/(2*(a*e - b*d)^2*(d + e*x)^(2*p + 3)*(3*p + 2*p^2 + 1)))

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(-3-2*p)*(g*x+f)*(b**2*x**2+2*a*b*x+a**2)**p,x)

[Out]

Exception raised: HeuristicGCDFailed

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