### 3.1251 $$\int \frac{(b d+2 c d x)^5}{(a+b x+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=84 $\frac{256}{3} c^2 d^5 \sqrt{a+b x+c x^2}-\frac{32 c d^5 (b+2 c x)^2}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}$

[Out]

(-2*d^5*(b + 2*c*x)^4)/(3*(a + b*x + c*x^2)^(3/2)) - (32*c*d^5*(b + 2*c*x)^2)/(3*Sqrt[a + b*x + c*x^2]) + (256
*c^2*d^5*Sqrt[a + b*x + c*x^2])/3

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Rubi [A]  time = 0.0439912, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {686, 629} $\frac{256}{3} c^2 d^5 \sqrt{a+b x+c x^2}-\frac{32 c d^5 (b+2 c x)^2}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*d^5*(b + 2*c*x)^4)/(3*(a + b*x + c*x^2)^(3/2)) - (32*c*d^5*(b + 2*c*x)^2)/(3*Sqrt[a + b*x + c*x^2]) + (256
*c^2*d^5*Sqrt[a + b*x + c*x^2])/3

Rule 686

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*(d + e*x)^(m - 1)*
(a + b*x + c*x^2)^(p + 1))/(b*(p + 1)), x] - Dist[(d*e*(m - 1))/(b*(p + 1)), 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] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2
*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 629

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

Rubi steps

\begin{align*} \int \frac{(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac{1}{3} \left (16 c d^2\right ) \int \frac{(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{32 c d^5 (b+2 c x)^2}{3 \sqrt{a+b x+c x^2}}+\frac{1}{3} \left (128 c^2 d^4\right ) \int \frac{b d+2 c d x}{\sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{32 c d^5 (b+2 c x)^2}{3 \sqrt{a+b x+c x^2}}+\frac{256}{3} c^2 d^5 \sqrt{a+b x+c x^2}\\ \end{align*}

Mathematica [A]  time = 0.0532253, size = 91, normalized size = 1.08 $\frac{d^5 \left (32 c^2 \left (8 a^2+12 a c x^2+3 c^2 x^4\right )+16 b^2 c \left (3 c x^2-2 a\right )+192 b c^2 x \left (2 a+c x^2\right )-48 b^3 c x-2 b^4\right )}{3 (a+x (b+c x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d^5*(-2*b^4 - 48*b^3*c*x + 192*b*c^2*x*(2*a + c*x^2) + 16*b^2*c*(-2*a + 3*c*x^2) + 32*c^2*(8*a^2 + 12*a*c*x^2
+ 3*c^2*x^4)))/(3*(a + x*(b + c*x))^(3/2))

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Maple [A]  time = 0.046, size = 91, normalized size = 1.1 \begin{align*}{\frac{2\,{d}^{5} \left ( 48\,{c}^{4}{x}^{4}+96\,b{c}^{3}{x}^{3}+192\,a{c}^{3}{x}^{2}+24\,{b}^{2}{c}^{2}{x}^{2}+192\,ab{c}^{2}x-24\,{b}^{3}cx+128\,{a}^{2}{c}^{2}-16\,ac{b}^{2}-{b}^{4} \right ) }{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*d^5*(48*c^4*x^4+96*b*c^3*x^3+192*a*c^3*x^2+24*b^2*c^2*x^2+192*a*b*c^2*x-24*b^3*c*x+128*a^2*c^2-16*a*b^2*c-
b^4)/(c*x^2+b*x+a)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 8.91066, size = 302, normalized size = 3.6 \begin{align*} \frac{2 \,{\left (48 \, c^{4} d^{5} x^{4} + 96 \, b c^{3} d^{5} x^{3} + 24 \,{\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d^{5} x^{2} - 24 \,{\left (b^{3} c - 8 \, a b c^{2}\right )} d^{5} x -{\left (b^{4} + 16 \, a b^{2} c - 128 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(48*c^4*d^5*x^4 + 96*b*c^3*d^5*x^3 + 24*(b^2*c^2 + 8*a*c^3)*d^5*x^2 - 24*(b^3*c - 8*a*b*c^2)*d^5*x - (b^4
+ 16*a*b^2*c - 128*a^2*c^2)*d^5)*sqrt(c*x^2 + b*x + a)/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^
2)

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Sympy [B]  time = 2.62557, size = 615, normalized size = 7.32 \begin{align*} \frac{256 a^{2} c^{2} d^{5}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{32 a b^{2} c d^{5}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{384 a b c^{2} d^{5} x}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{384 a c^{3} d^{5} x^{2}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{2 b^{4} d^{5}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{48 b^{3} c d^{5} x}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{48 b^{2} c^{2} d^{5} x^{2}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{192 b c^{3} d^{5} x^{3}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{96 c^{4} d^{5} x^{4}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

256*a**2*c**2*d**5/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2
)) - 32*a*b**2*c*d**5/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x
**2)) + 384*a*b*c**2*d**5*x/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x
+ c*x**2)) + 384*a*c**3*d**5*x**2/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(
a + b*x + c*x**2)) - 2*b**4*d**5/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a
+ b*x + c*x**2)) - 48*b**3*c*d**5*x/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt
(a + b*x + c*x**2)) + 48*b**2*c**2*d**5*x**2/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*
x**2*sqrt(a + b*x + c*x**2)) + 192*b*c**3*d**5*x**3/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2)
+ 3*c*x**2*sqrt(a + b*x + c*x**2)) + 96*c**4*d**5*x**4/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x
**2) + 3*c*x**2*sqrt(a + b*x + c*x**2))

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Giac [B]  time = 1.15636, size = 521, normalized size = 6.2 \begin{align*} \frac{2 \,{\left (24 \,{\left ({\left (2 \,{\left (\frac{{\left (b^{4} c^{6} d^{5} - 8 \, a b^{2} c^{7} d^{5} + 16 \, a^{2} c^{8} d^{5}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{2 \,{\left (b^{5} c^{5} d^{5} - 8 \, a b^{3} c^{6} d^{5} + 16 \, a^{2} b c^{7} d^{5}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{b^{6} c^{4} d^{5} - 48 \, a^{2} b^{2} c^{6} d^{5} + 128 \, a^{3} c^{7} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{7} c^{3} d^{5} - 16 \, a b^{5} c^{4} d^{5} + 80 \, a^{2} b^{3} c^{5} d^{5} - 128 \, a^{3} b c^{6} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{8} c^{2} d^{5} + 8 \, a b^{6} c^{3} d^{5} - 240 \, a^{2} b^{4} c^{4} d^{5} + 1280 \, a^{3} b^{2} c^{5} d^{5} - 2048 \, a^{4} c^{6} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(24*((2*((b^4*c^6*d^5 - 8*a*b^2*c^7*d^5 + 16*a^2*c^8*d^5)*x/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 2*(b^5*
c^5*d^5 - 8*a*b^3*c^6*d^5 + 16*a^2*b*c^7*d^5)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + (b^6*c^4*d^5 - 48*a^2*
b^2*c^6*d^5 + 128*a^3*c^7*d^5)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x - (b^7*c^3*d^5 - 16*a*b^5*c^4*d^5 + 80*
a^2*b^3*c^5*d^5 - 128*a^3*b*c^6*d^5)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x - (b^8*c^2*d^5 + 8*a*b^6*c^3*d^5
- 240*a^2*b^4*c^4*d^5 + 1280*a^3*b^2*c^5*d^5 - 2048*a^4*c^6*d^5)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2
+ b*x + a)^(3/2)