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

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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]

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]

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 = 0.05, size = 91, normalized size = 1.08 \begin {gather*} \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 )-2 b^4-48 b^3 c x\right )}{3 (a+x (b+c x))^{3/2}} \end {gather*}

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))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.33, size = 117, normalized size = 1.39 \begin {gather*} -\frac {2 \left (-128 a^2 c^2 d^5+16 a b^2 c d^5-192 a b c^2 d^5 x-192 a c^3 d^5 x^2+b^4 d^5+24 b^3 c d^5 x-24 b^2 c^2 d^5 x^2-96 b c^3 d^5 x^3-48 c^4 d^5 x^4\right )}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b^4*d^5 + 16*a*b^2*c*d^5 - 128*a^2*c^2*d^5 + 24*b^3*c*d^5*x - 192*a*b*c^2*d^5*x - 24*b^2*c^2*d^5*x^2 - 19
2*a*c^3*d^5*x^2 - 96*b*c^3*d^5*x^3 - 48*c^4*d^5*x^4))/(3*(a + b*x + c*x^2)^(3/2))

________________________________________________________________________________________

fricas [A]  time = 0.91, size = 140, normalized size = 1.67 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

giac [B]  time = 0.39, size = 386, normalized size = 4.60 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 91, normalized size = 1.08 \begin {gather*} \frac {2 \left (48 c^{4} x^{4}+96 b \,c^{3} x^{3}+192 a \,c^{3} x^{2}+24 x^{2} b^{2} c^{2}+192 a b \,c^{2} x -24 x \,b^{3} c +128 a^{2} c^{2}-16 a \,b^{2} c -b^{4}\right ) d^{5}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}} \end {gather*}

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)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

________________________________________________________________________________________

mupad [B]  time = 0.91, size = 118, normalized size = 1.40 \begin {gather*} -\frac {2\,b^4\,d^5+32\,a^2\,c^2\,d^5-96\,c^2\,d^5\,{\left (c\,x^2+b\,x+a\right )}^2-16\,a\,b^2\,c\,d^5-192\,a\,c^2\,d^5\,\left (c\,x^2+b\,x+a\right )+48\,b^2\,c\,d^5\,\left (c\,x^2+b\,x+a\right )}{\sqrt {c\,x^2+b\,x+a}\,\left (3\,c\,x^2+3\,b\,x+3\,a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b^4*d^5 + 32*a^2*c^2*d^5 - 96*c^2*d^5*(a + b*x + c*x^2)^2 - 16*a*b^2*c*d^5 - 192*a*c^2*d^5*(a + b*x + c*x^
2) + 48*b^2*c*d^5*(a + b*x + c*x^2))/((a + b*x + c*x^2)^(1/2)*(3*a + 3*b*x + 3*c*x^2))

________________________________________________________________________________________

sympy [B]  time = 1.97, size = 615, normalized size = 7.32 \begin {gather*} \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 {gather*}

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))

________________________________________________________________________________________