\(\int \frac {(b d+2 c d x)^5}{(a+b x+c x^2)^{5/2}} \, dx\) [1251]

Optimal result

Integrand size = 26, antiderivative size = 84 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {32 c d^5 (b+2 c x)^2}{3 \sqrt {a+b x+c x^2}}+\frac {256}{3} c^2 d^5 \sqrt {a+b x+c x^2} \]

[Out]

-2/3*d^5*(2*c*x+b)^4/(c*x^2+b*x+a)^(3/2)-32/3*c*d^5*(2*c*x+b)^2/(c*x^2+b*x+a)^(1/2)+256/3*c^2*d^5*(c*x^2+b*x+a
)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {700, 643} \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {256}{3} c^2 d^5 \sqrt {a+b x+c x^2}-\frac {32 c d^5 (b+2 c x)^2}{3 \sqrt {a+b x+c x^2}}-\frac {2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[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 643

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 700

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*} \text {integral}& = -\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] (verified)

Time = 0.77 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.08 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {d^5 \left (-2 b^4-48 b^3 c x+192 b c^2 x \left (2 a+c x^2\right )+16 b^2 c \left (-2 a+3 c x^2\right )+32 c^2 \left (8 a^2+12 a c x^2+3 c^2 x^4\right )\right )}{3 (a+x (b+c x))^{3/2}} \]

[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))

Maple [A] (verified)

Time = 2.85 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83

method	result	size
risch	\(32 c^{2} d^{5} \sqrt {c \,x^{2}+b x +a}+\frac {2 \left (24 c^{2} x^{2}+24 b c x +20 a c +b^{2}\right ) \left (4 a c -b^{2}\right ) d^{5}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\)	\(70\)
pseudoelliptic	\(\frac {256 \left (\frac {3 c^{4} x^{4}}{8}+\frac {3 x^{2} \left (\frac {b x}{2}+a \right ) c^{3}}{2}+\left (\frac {3}{16} b^{2} x^{2}+\frac {3}{2} a b x +a^{2}\right ) c^{2}-\frac {\left (\frac {3 b x}{2}+a \right ) b^{2} c}{8}-\frac {b^{4}}{128}\right ) d^{5}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\)	\(79\)
gosper	\(\frac {2 d^{5} \left (48 c^{4} x^{4}+96 b \,c^{3} x^{3}+192 x^{2} c^{3} a +24 b^{2} c^{2} x^{2}+192 a b \,c^{2} x -24 b^{3} c x +128 a^{2} c^{2}-16 a \,b^{2} c -b^{4}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\)	\(91\)
trager	\(\frac {2 d^{5} \left (48 c^{4} x^{4}+96 b \,c^{3} x^{3}+192 x^{2} c^{3} a +24 b^{2} c^{2} x^{2}+192 a b \,c^{2} x -24 b^{3} c x +128 a^{2} c^{2}-16 a \,b^{2} c -b^{4}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\)	\(91\)
default	\(\text {Expression too large to display}\)	\(1936\)

[In]

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

[Out]

32*c^2*d^5*(c*x^2+b*x+a)^(1/2)+2/3*(24*c^2*x^2+24*b*c*x+20*a*c+b^2)*(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*d^5

Fricas [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.67 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (48 \, c^{4} d^{5} x^{4} + 96 \, b c^{3} d^{5} x^{3} + 24 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d^{5} x^{2} - 24 \, {\left (b^{3} c - 8 \, a b c^{2}\right )} d^{5} x - {\left (b^{4} + 16 \, a b^{2} c - 128 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 702 vs. \(2 (82) = 164\).

Time = 0.60 (sec) , antiderivative size = 702, normalized size of antiderivative = 8.36 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\begin {cases} \frac {256 a^{2} c^{2} d^{5}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} - \frac {32 a b^{2} c d^{5}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} + \frac {384 a b c^{2} d^{5} x}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} + \frac {384 a c^{3} d^{5} x^{2}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} - \frac {2 b^{4} d^{5}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} - \frac {48 b^{3} c d^{5} x}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} + \frac {48 b^{2} c^{2} d^{5} x^{2}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} + \frac {192 b c^{3} d^{5} x^{3}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} + \frac {96 c^{4} d^{5} x^{4}}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} & \text {for}\: a \neq - x \left (b + c x\right ) \\\tilde {\infty } b^{5} d^{5} x + \tilde {\infty } b^{4} c d^{5} x^{2} + \tilde {\infty } b^{3} c^{2} d^{5} x^{3} + \tilde {\infty } b^{2} c^{3} d^{5} x^{4} + \tilde {\infty } b c^{4} d^{5} x^{5} + \tilde {\infty } c^{5} d^{5} x^{6} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((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*s
qrt(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)), Ne(a, -x*(b + c*x))), (zoo*b**5*d**5*x + zoo*b**4*c*d**5*x
**2 + zoo*b**3*c**2*d**5*x**3 + zoo*b**2*c**3*d**5*x**4 + zoo*b*c**4*d**5*x**5 + zoo*c**5*d**5*x**6, True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[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 deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (72) = 144\).

Time = 0.29 (sec) , antiderivative size = 386, normalized size of antiderivative = 4.60 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (24 \, {\left ({\left (2 \, {\left (\frac {{\left (b^{4} c^{6} d^{5} - 8 \, a b^{2} c^{7} d^{5} + 16 \, a^{2} c^{8} d^{5}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac {2 \, {\left (b^{5} c^{5} d^{5} - 8 \, a b^{3} c^{6} d^{5} + 16 \, a^{2} b c^{7} d^{5}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac {b^{6} c^{4} d^{5} - 48 \, a^{2} b^{2} c^{6} d^{5} + 128 \, a^{3} c^{7} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac {b^{7} c^{3} d^{5} - 16 \, a b^{5} c^{4} d^{5} + 80 \, a^{2} b^{3} c^{5} d^{5} - 128 \, a^{3} b c^{6} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac {b^{8} c^{2} d^{5} + 8 \, a b^{6} c^{3} d^{5} - 240 \, a^{2} b^{4} c^{4} d^{5} + 1280 \, a^{3} b^{2} c^{5} d^{5} - 2048 \, a^{4} c^{6} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]

[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)

Mupad [B] (verification not implemented)

Time = 9.70 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.40 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2\,b^4\,d^5+32\,a^2\,c^2\,d^5-96\,c^2\,d^5\,{\left (c\,x^2+b\,x+a\right )}^2-16\,a\,b^2\,c\,d^5-192\,a\,c^2\,d^5\,\left (c\,x^2+b\,x+a\right )+48\,b^2\,c\,d^5\,\left (c\,x^2+b\,x+a\right )}{\sqrt {c\,x^2+b\,x+a}\,\left (3\,c\,x^2+3\,b\,x+3\,a\right )} \]

[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))