3.1259 \(\int \frac{1}{(b d+2 c d x)^4 (a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=162 \[ \frac{512 c^2 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^4 (b+2 c x)}+\frac{256 c^2 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3}+\frac{16 c}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}-\frac{2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

-2/(3*(b^2 - 4*a*c)*d^4*(b + 2*c*x)^3*(a + b*x + c*x^2)^(3/2)) + (16*c)/((b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^3*Sqr
t[a + b*x + c*x^2]) + (256*c^2*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3) + (512*c^2*Sqrt[a
+ b*x + c*x^2])/(3*(b^2 - 4*a*c)^4*d^4*(b + 2*c*x))

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Rubi [A]  time = 0.0794517, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {687, 693, 682} \[ \frac{512 c^2 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^4 (b+2 c x)}+\frac{256 c^2 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3}+\frac{16 c}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}-\frac{2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2)),x]

[Out]

-2/(3*(b^2 - 4*a*c)*d^4*(b + 2*c*x)^3*(a + b*x + c*x^2)^(3/2)) + (16*c)/((b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^3*Sqr
t[a + b*x + c*x^2]) + (256*c^2*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3) + (512*c^2*Sqrt[a
+ b*x + c*x^2])/(3*(b^2 - 4*a*c)^4*d^4*(b + 2*c*x))

Rule 687

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

Rule 693

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

Rule 682

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

Rubi steps

\begin{align*} \int \frac{1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}-\frac{(8 c) \int \frac{1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx}{b^2-4 a c}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}+\frac{16 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}+\frac{\left (128 c^2\right ) \int \frac{1}{(b d+2 c d x)^4 \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}+\frac{16 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}+\frac{256 c^2 \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac{\left (256 c^2\right ) \int \frac{1}{(b d+2 c d x)^2 \sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^3 d^2}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}+\frac{16 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}+\frac{256 c^2 \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac{512 c^2 \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x)}\\ \end{align*}

Mathematica [A]  time = 0.0844971, size = 178, normalized size = 1.1 \[ \frac{2 \left (48 b^2 c^2 \left (3 a^2+44 a c x^2+72 c^2 x^4\right )+384 b c^3 x \left (a^2+8 a c x^2+8 c^2 x^4\right )+64 c^3 \left (6 a^2 c x^2-a^3+24 a c^2 x^4+16 c^3 x^6\right )+64 b^3 c^2 x \left (9 a+28 c x^2\right )+12 b^4 c \left (3 a+34 c x^2\right )+24 b^5 c x-b^6\right )}{3 d^4 \left (b^2-4 a c\right )^4 (b+2 c x)^3 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*(-b^6 + 24*b^5*c*x + 64*b^3*c^2*x*(9*a + 28*c*x^2) + 12*b^4*c*(3*a + 34*c*x^2) + 384*b*c^3*x*(a^2 + 8*a*c*x
^2 + 8*c^2*x^4) + 48*b^2*c^2*(3*a^2 + 44*a*c*x^2 + 72*c^2*x^4) + 64*c^3*(-a^3 + 6*a^2*c*x^2 + 24*a*c^2*x^4 + 1
6*c^3*x^6)))/(3*(b^2 - 4*a*c)^4*d^4*(b + 2*c*x)^3*(a + x*(b + c*x))^(3/2))

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Maple [A]  time = 0.052, size = 218, normalized size = 1.4 \begin{align*} -{\frac{-2048\,{c}^{6}{x}^{6}-6144\,b{c}^{5}{x}^{5}-3072\,a{c}^{5}{x}^{4}-6912\,{b}^{2}{c}^{4}{x}^{4}-6144\,ab{c}^{4}{x}^{3}-3584\,{b}^{3}{c}^{3}{x}^{3}-768\,{a}^{2}{c}^{4}{x}^{2}-4224\,a{b}^{2}{c}^{3}{x}^{2}-816\,{b}^{4}{c}^{2}{x}^{2}-768\,{a}^{2}b{c}^{3}x-1152\,a{b}^{3}{c}^{2}x-48\,{b}^{5}cx+128\,{a}^{3}{c}^{3}-288\,{a}^{2}{b}^{2}{c}^{2}-72\,a{b}^{4}c+2\,{b}^{6}}{3\, \left ( 256\,{a}^{4}{c}^{4}-256\,{a}^{3}{b}^{2}{c}^{3}+96\,{a}^{2}{b}^{4}{c}^{2}-16\,a{b}^{6}c+{b}^{8} \right ){d}^{4} \left ( 2\,cx+b \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(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x)

[Out]

-2/3*(-1024*c^6*x^6-3072*b*c^5*x^5-1536*a*c^5*x^4-3456*b^2*c^4*x^4-3072*a*b*c^4*x^3-1792*b^3*c^3*x^3-384*a^2*c
^4*x^2-2112*a*b^2*c^3*x^2-408*b^4*c^2*x^2-384*a^2*b*c^3*x-576*a*b^3*c^2*x-24*b^5*c*x+64*a^3*c^3-144*a^2*b^2*c^
2-36*a*b^4*c+b^6)/(2*c*x+b)^3/d^4/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(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(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 74.7853, size = 1494, normalized size = 9.22 \begin{align*} \frac{2 \,{\left (1024 \, c^{6} x^{6} + 3072 \, b c^{5} x^{5} - b^{6} + 36 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 384 \,{\left (9 \, b^{2} c^{4} + 4 \, a c^{5}\right )} x^{4} + 256 \,{\left (7 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} x^{3} + 24 \,{\left (17 \, b^{4} c^{2} + 88 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 24 \,{\left (b^{5} c + 24 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (8 \,{\left (b^{8} c^{5} - 16 \, a b^{6} c^{6} + 96 \, a^{2} b^{4} c^{7} - 256 \, a^{3} b^{2} c^{8} + 256 \, a^{4} c^{9}\right )} d^{4} x^{7} + 28 \,{\left (b^{9} c^{4} - 16 \, a b^{7} c^{5} + 96 \, a^{2} b^{5} c^{6} - 256 \, a^{3} b^{3} c^{7} + 256 \, a^{4} b c^{8}\right )} d^{4} x^{6} + 2 \,{\left (19 \, b^{10} c^{3} - 296 \, a b^{8} c^{4} + 1696 \, a^{2} b^{6} c^{5} - 4096 \, a^{3} b^{4} c^{6} + 2816 \, a^{4} b^{2} c^{7} + 2048 \, a^{5} c^{8}\right )} d^{4} x^{5} + 5 \,{\left (5 \, b^{11} c^{2} - 72 \, a b^{9} c^{3} + 352 \, a^{2} b^{7} c^{4} - 512 \, a^{3} b^{5} c^{5} - 768 \, a^{4} b^{3} c^{6} + 2048 \, a^{5} b c^{7}\right )} d^{4} x^{4} + 4 \,{\left (2 \, b^{12} c - 23 \, a b^{10} c^{2} + 50 \, a^{2} b^{8} c^{3} + 320 \, a^{3} b^{6} c^{4} - 1600 \, a^{4} b^{4} c^{5} + 1792 \, a^{5} b^{2} c^{6} + 512 \, a^{6} c^{7}\right )} d^{4} x^{3} +{\left (b^{13} - 2 \, a b^{11} c - 116 \, a^{2} b^{9} c^{2} + 896 \, a^{3} b^{7} c^{3} - 2176 \, a^{4} b^{5} c^{4} + 512 \, a^{5} b^{3} c^{5} + 3072 \, a^{6} b c^{6}\right )} d^{4} x^{2} + 2 \,{\left (a b^{12} - 13 \, a^{2} b^{10} c + 48 \, a^{3} b^{8} c^{2} + 32 \, a^{4} b^{6} c^{3} - 512 \, a^{5} b^{4} c^{4} + 768 \, a^{6} b^{2} c^{5}\right )} d^{4} x +{\left (a^{2} b^{11} - 16 \, a^{3} b^{9} c + 96 \, a^{4} b^{7} c^{2} - 256 \, a^{5} b^{5} c^{3} + 256 \, a^{6} b^{3} c^{4}\right )} d^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

2/3*(1024*c^6*x^6 + 3072*b*c^5*x^5 - b^6 + 36*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3 + 384*(9*b^2*c^4 + 4*a*c^
5)*x^4 + 256*(7*b^3*c^3 + 12*a*b*c^4)*x^3 + 24*(17*b^4*c^2 + 88*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 24*(b^5*c + 24*a
*b^3*c^2 + 16*a^2*b*c^3)*x)*sqrt(c*x^2 + b*x + a)/(8*(b^8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^
8 + 256*a^4*c^9)*d^4*x^7 + 28*(b^9*c^4 - 16*a*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*
x^6 + 2*(19*b^10*c^3 - 296*a*b^8*c^4 + 1696*a^2*b^6*c^5 - 4096*a^3*b^4*c^6 + 2816*a^4*b^2*c^7 + 2048*a^5*c^8)*
d^4*x^5 + 5*(5*b^11*c^2 - 72*a*b^9*c^3 + 352*a^2*b^7*c^4 - 512*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)
*d^4*x^4 + 4*(2*b^12*c - 23*a*b^10*c^2 + 50*a^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600*a^4*b^4*c^5 + 1792*a^5*b^2*c^
6 + 512*a^6*c^7)*d^4*x^3 + (b^13 - 2*a*b^11*c - 116*a^2*b^9*c^2 + 896*a^3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5
*b^3*c^5 + 3072*a^6*b*c^6)*d^4*x^2 + 2*(a*b^12 - 13*a^2*b^10*c + 48*a^3*b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4
*c^4 + 768*a^6*b^2*c^5)*d^4*x + (a^2*b^11 - 16*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5*c^3 + 256*a^6*b^3*c^4)
*d^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{2} b^{4} \sqrt{a + b x + c x^{2}} + 8 a^{2} b^{3} c x \sqrt{a + b x + c x^{2}} + 24 a^{2} b^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 32 a^{2} b c^{3} x^{3} \sqrt{a + b x + c x^{2}} + 16 a^{2} c^{4} x^{4} \sqrt{a + b x + c x^{2}} + 2 a b^{5} x \sqrt{a + b x + c x^{2}} + 18 a b^{4} c x^{2} \sqrt{a + b x + c x^{2}} + 64 a b^{3} c^{2} x^{3} \sqrt{a + b x + c x^{2}} + 112 a b^{2} c^{3} x^{4} \sqrt{a + b x + c x^{2}} + 96 a b c^{4} x^{5} \sqrt{a + b x + c x^{2}} + 32 a c^{5} x^{6} \sqrt{a + b x + c x^{2}} + b^{6} x^{2} \sqrt{a + b x + c x^{2}} + 10 b^{5} c x^{3} \sqrt{a + b x + c x^{2}} + 41 b^{4} c^{2} x^{4} \sqrt{a + b x + c x^{2}} + 88 b^{3} c^{3} x^{5} \sqrt{a + b x + c x^{2}} + 104 b^{2} c^{4} x^{6} \sqrt{a + b x + c x^{2}} + 64 b c^{5} x^{7} \sqrt{a + b x + c x^{2}} + 16 c^{6} x^{8} \sqrt{a + b x + c x^{2}}}\, dx}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/(a**2*b**4*sqrt(a + b*x + c*x**2) + 8*a**2*b**3*c*x*sqrt(a + b*x + c*x**2) + 24*a**2*b**2*c**2*x**2
*sqrt(a + b*x + c*x**2) + 32*a**2*b*c**3*x**3*sqrt(a + b*x + c*x**2) + 16*a**2*c**4*x**4*sqrt(a + b*x + c*x**2
) + 2*a*b**5*x*sqrt(a + b*x + c*x**2) + 18*a*b**4*c*x**2*sqrt(a + b*x + c*x**2) + 64*a*b**3*c**2*x**3*sqrt(a +
b*x + c*x**2) + 112*a*b**2*c**3*x**4*sqrt(a + b*x + c*x**2) + 96*a*b*c**4*x**5*sqrt(a + b*x + c*x**2) + 32*a*
c**5*x**6*sqrt(a + b*x + c*x**2) + b**6*x**2*sqrt(a + b*x + c*x**2) + 10*b**5*c*x**3*sqrt(a + b*x + c*x**2) +
41*b**4*c**2*x**4*sqrt(a + b*x + c*x**2) + 88*b**3*c**3*x**5*sqrt(a + b*x + c*x**2) + 104*b**2*c**4*x**6*sqrt(
a + b*x + c*x**2) + 64*b*c**5*x**7*sqrt(a + b*x + c*x**2) + 16*c**6*x**8*sqrt(a + b*x + c*x**2)), x)/d**4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError