3.2351 \(\int \frac{(a+b x+c x^2)^{3/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=225 \[ -\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-3*(b^2 - 4*a*c)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(64*(c*d^2 - b*d*e + a*e^2)^2*(d + e*
x)^2) + ((b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(8*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) + (3
*(b^2 - 4*a*c)^2*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])
])/(128*(c*d^2 - b*d*e + a*e^2)^(5/2))

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Rubi [A]  time = 0.150275, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {720, 724, 206} \[ -\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

(-3*(b^2 - 4*a*c)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(64*(c*d^2 - b*d*e + a*e^2)^2*(d + e*
x)^2) + ((b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(8*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) + (3
*(b^2 - 4*a*c)^2*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])
])/(128*(c*d^2 - b*d*e + a*e^2)^(5/2))

Rule 720

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

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx &=\frac{(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{8 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac{\left (3 \left (b^2-4 a c\right )\right ) \int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx}{16 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{3 \left (b^2-4 a c\right ) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{8 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac{\left (3 \left (b^2-4 a c\right )^2\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{128 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{3 \left (b^2-4 a c\right ) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{8 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac{\left (3 \left (b^2-4 a c\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{64 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{3 \left (b^2-4 a c\right ) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{8 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{128 \left (c d^2-b d e+a e^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.768824, size = 222, normalized size = 0.99 \[ -\frac{3 \left (b^2-4 a c\right ) \left (\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left (e (a e-b d)+c d^2\right )}\right )+\frac{2 (a+x (b+c x))^{3/2} (2 a e-b d+b e x-2 c d x)}{(d+e x)^4}}{16 \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

-((2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 + 3*(b^2 - 4*a*c)*((Sqrt[a + x*(b
+ c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh
[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(
-(b*d) + a*e))^(3/2))))/(16*(c*d^2 + e*(-(b*d) + a*e)))

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Maple [B]  time = 0.24, size = 15932, normalized size = 70.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)/(e*x+d)^5,x)

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**5,x)

[Out]

Timed out

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Giac [B]  time = 18.765, size = 3803, normalized size = 16.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

1/128*(2*sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2)
*(2*(4*(3*(2*c^4*d^7*e^19*sgn(1/(x*e + d)) - 7*b*c^3*d^6*e^20*sgn(1/(x*e + d)) + 9*b^2*c^2*d^5*e^21*sgn(1/(x*e
+ d)) + 6*a*c^3*d^5*e^21*sgn(1/(x*e + d)) - 5*b^3*c*d^4*e^22*sgn(1/(x*e + d)) - 15*a*b*c^2*d^4*e^22*sgn(1/(x*
e + d)) + b^4*d^3*e^23*sgn(1/(x*e + d)) + 12*a*b^2*c*d^3*e^23*sgn(1/(x*e + d)) + 6*a^2*c^2*d^3*e^23*sgn(1/(x*e
+ d)) - 3*a*b^3*d^2*e^24*sgn(1/(x*e + d)) - 9*a^2*b*c*d^2*e^24*sgn(1/(x*e + d)) + 3*a^2*b^2*d*e^25*sgn(1/(x*e
+ d)) + 2*a^3*c*d*e^25*sgn(1/(x*e + d)) - a^3*b*e^26*sgn(1/(x*e + d)))/(c^4*d^8*e^8 - 4*b*c^3*d^7*e^9 + 6*b^2
*c^2*d^6*e^10 + 4*a*c^3*d^6*e^10 - 4*b^3*c*d^5*e^11 - 12*a*b*c^2*d^5*e^11 + b^4*d^4*e^12 + 12*a*b^2*c*d^4*e^12
+ 6*a^2*c^2*d^4*e^12 - 4*a*b^3*d^3*e^13 - 12*a^2*b*c*d^3*e^13 + 6*a^2*b^2*d^2*e^14 + 4*a^3*c*d^2*e^14 - 4*a^3
*b*d*e^15 + a^4*e^16) - 2*(c^4*d^8*e^20*sgn(1/(x*e + d)) - 4*b*c^3*d^7*e^21*sgn(1/(x*e + d)) + 6*b^2*c^2*d^6*e
^22*sgn(1/(x*e + d)) + 4*a*c^3*d^6*e^22*sgn(1/(x*e + d)) - 4*b^3*c*d^5*e^23*sgn(1/(x*e + d)) - 12*a*b*c^2*d^5*
e^23*sgn(1/(x*e + d)) + b^4*d^4*e^24*sgn(1/(x*e + d)) + 12*a*b^2*c*d^4*e^24*sgn(1/(x*e + d)) + 6*a^2*c^2*d^4*e
^24*sgn(1/(x*e + d)) - 4*a*b^3*d^3*e^25*sgn(1/(x*e + d)) - 12*a^2*b*c*d^3*e^25*sgn(1/(x*e + d)) + 6*a^2*b^2*d^
2*e^26*sgn(1/(x*e + d)) + 4*a^3*c*d^2*e^26*sgn(1/(x*e + d)) - 4*a^3*b*d*e^27*sgn(1/(x*e + d)) + a^4*e^28*sgn(1
/(x*e + d)))*e^(-1)/((c^4*d^8*e^8 - 4*b*c^3*d^7*e^9 + 6*b^2*c^2*d^6*e^10 + 4*a*c^3*d^6*e^10 - 4*b^3*c*d^5*e^11
- 12*a*b*c^2*d^5*e^11 + b^4*d^4*e^12 + 12*a*b^2*c*d^4*e^12 + 6*a^2*c^2*d^4*e^12 - 4*a*b^3*d^3*e^13 - 12*a^2*b
*c*d^3*e^13 + 6*a^2*b^2*d^2*e^14 + 4*a^3*c*d^2*e^14 - 4*a^3*b*d*e^15 + a^4*e^16)*(x*e + d)))*e^(-1)/(x*e + d)
- (24*c^4*d^6*e^18*sgn(1/(x*e + d)) - 72*b*c^3*d^5*e^19*sgn(1/(x*e + d)) + 73*b^2*c^2*d^4*e^20*sgn(1/(x*e + d)
) + 68*a*c^3*d^4*e^20*sgn(1/(x*e + d)) - 26*b^3*c*d^3*e^21*sgn(1/(x*e + d)) - 136*a*b*c^2*d^3*e^21*sgn(1/(x*e
+ d)) + b^4*d^2*e^22*sgn(1/(x*e + d)) + 70*a*b^2*c*d^2*e^22*sgn(1/(x*e + d)) + 64*a^2*c^2*d^2*e^22*sgn(1/(x*e
+ d)) - 2*a*b^3*d*e^23*sgn(1/(x*e + d)) - 64*a^2*b*c*d*e^23*sgn(1/(x*e + d)) + a^2*b^2*e^24*sgn(1/(x*e + d)) +
20*a^3*c*e^24*sgn(1/(x*e + d)))/(c^4*d^8*e^8 - 4*b*c^3*d^7*e^9 + 6*b^2*c^2*d^6*e^10 + 4*a*c^3*d^6*e^10 - 4*b^
3*c*d^5*e^11 - 12*a*b*c^2*d^5*e^11 + b^4*d^4*e^12 + 12*a*b^2*c*d^4*e^12 + 6*a^2*c^2*d^4*e^12 - 4*a*b^3*d^3*e^1
3 - 12*a^2*b*c*d^3*e^13 + 6*a^2*b^2*d^2*e^14 + 4*a^3*c*d^2*e^14 - 4*a^3*b*d*e^15 + a^4*e^16))*e^(-1)/(x*e + d)
+ (16*c^4*d^5*e^17*sgn(1/(x*e + d)) - 40*b*c^3*d^4*e^18*sgn(1/(x*e + d)) + 26*b^2*c^2*d^3*e^19*sgn(1/(x*e + d
)) + 56*a*c^3*d^3*e^19*sgn(1/(x*e + d)) + b^3*c*d^2*e^20*sgn(1/(x*e + d)) - 84*a*b*c^2*d^2*e^20*sgn(1/(x*e + d
)) - 3*b^4*d*e^21*sgn(1/(x*e + d)) + 22*a*b^2*c*d*e^21*sgn(1/(x*e + d)) + 40*a^2*c^2*d*e^21*sgn(1/(x*e + d)) +
3*a*b^3*e^22*sgn(1/(x*e + d)) - 20*a^2*b*c*e^22*sgn(1/(x*e + d)))/(c^4*d^8*e^8 - 4*b*c^3*d^7*e^9 + 6*b^2*c^2*
d^6*e^10 + 4*a*c^3*d^6*e^10 - 4*b^3*c*d^5*e^11 - 12*a*b*c^2*d^5*e^11 + b^4*d^4*e^12 + 12*a*b^2*c*d^4*e^12 + 6*
a^2*c^2*d^4*e^12 - 4*a*b^3*d^3*e^13 - 12*a^2*b*c*d^3*e^13 + 6*a^2*b^2*d^2*e^14 + 4*a^3*c*d^2*e^14 - 4*a^3*b*d*
e^15 + a^4*e^16)) - 3*(b^4*e^14*sgn(1/(x*e + d)) - 8*a*b^2*c*e^14*sgn(1/(x*e + d)) + 16*a^2*c^2*e^14*sgn(1/(x*
e + d)))*sqrt(c*d^2 - b*d*e + a*e^2)*log(abs(2*(c*d^2 - b*d*e + a*e^2)*(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e
+ d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2) + sqrt(c*d^2*e^2 - b*d*e^3 + a*e^4)*e^(-1)/(x*
e + d)) - sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d - b*e)))/(c^4*d^8*e - 4*b*c^3*d^7*e^2 + 6*b^2*c^2*d^6*e^3 + 4*a*c
^3*d^6*e^3 - 4*b^3*c*d^5*e^4 - 12*a*b*c^2*d^5*e^4 + b^4*d^4*e^5 + 12*a*b^2*c*d^4*e^5 + 6*a^2*c^2*d^4*e^5 - 4*a
*b^3*d^3*e^6 - 12*a^2*b*c*d^3*e^6 + 6*a^2*b^2*d^2*e^7 + 4*a^3*c*d^2*e^7 - 4*a^3*b*d*e^8 + a^4*e^9) - (32*c^(9/
2)*d^5*e^9 - 80*b*c^(7/2)*d^4*e^10 + 52*b^2*c^(5/2)*d^3*e^11 + 112*a*c^(7/2)*d^3*e^11 + 2*b^3*c^(3/2)*d^2*e^12
- 168*a*b*c^(5/2)*d^2*e^12 - 6*b^4*sqrt(c)*d*e^13 + 44*a*b^2*c^(3/2)*d*e^13 + 80*a^2*c^(5/2)*d*e^13 - 3*sqrt(
c*d^2 - b*d*e + a*e^2)*b^4*e^13*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d +
2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^2*c*e^13*log(abs(2*c^
(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2
)*b*e)) - 48*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*c^2*e^13*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 -
b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) + 6*a*b^3*sqrt(c)*e^14 - 40*a^2*b*c^
(3/2)*e^14)*sgn(1/(x*e + d))/(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3
- 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3
*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8))*e^2