3.1249 \(\int \frac{1}{(b d+2 c d x)^4 (a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=118 \[ -\frac{64 c \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)}-\frac{32 c \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3}-\frac{2}{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt{a+b x+c x^2}} \]

[Out]

-2/((b^2 - 4*a*c)*d^4*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2]) - (32*c*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^2*d
^4*(b + 2*c*x)^3) - (64*c*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x))

________________________________________________________________________________________

Rubi [A]  time = 0.0552308, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {687, 693, 682} \[ -\frac{64 c \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)}-\frac{32 c \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3}-\frac{2}{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x]

[Out]

-2/((b^2 - 4*a*c)*d^4*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2]) - (32*c*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^2*d
^4*(b + 2*c*x)^3) - (64*c*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^3*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 )^{3/2}} \, dx &=-\frac{2}{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}-\frac{(16 c) \int \frac{1}{(b d+2 c d x)^4 \sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}-\frac{32 c \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3}-\frac{(32 c) \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 )^2 d^2}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}-\frac{32 c \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3}-\frac{64 c \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)}\\ \end{align*}

Mathematica [A]  time = 0.0502315, size = 108, normalized size = 0.92 \[ -\frac{2 \left (16 c^2 \left (-a^2+4 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (3 a+22 c x^2\right )+64 b c^2 x \left (a+4 c x^2\right )+48 b^3 c x+3 b^4\right )}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(3*b^4 + 48*b^3*c*x + 64*b*c^2*x*(a + 4*c*x^2) + 8*b^2*c*(3*a + 22*c*x^2) + 16*c^2*(-a^2 + 4*a*c*x^2 + 8*c
^2*x^4)))/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3*Sqrt[a + x*(b + c*x)])

________________________________________________________________________________________

Maple [A]  time = 0.048, size = 133, normalized size = 1.1 \begin{align*} -{\frac{-256\,{c}^{4}{x}^{4}-512\,b{c}^{3}{x}^{3}-128\,a{c}^{3}{x}^{2}-352\,{b}^{2}{c}^{2}{x}^{2}-128\,ab{c}^{2}x-96\,{b}^{3}cx+32\,{a}^{2}{c}^{2}-48\,ac{b}^{2}-6\,{b}^{4}}{3\, \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){d}^{4} \left ( 2\,cx+b \right ) ^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \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)^(3/2),x)

[Out]

-2/3*(-128*c^4*x^4-256*b*c^3*x^3-64*a*c^3*x^2-176*b^2*c^2*x^2-64*a*b*c^2*x-48*b^3*c*x+16*a^2*c^2-24*a*b^2*c-3*
b^4)/(2*c*x+b)^3/d^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(c*x^2+b*x+a)^(1/2)

________________________________________________________________________________________

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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 27.7697, size = 824, normalized size = 6.98 \begin{align*} -\frac{2 \,{\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} + 3 \, b^{4} + 24 \, a b^{2} c - 16 \, a^{2} c^{2} + 16 \,{\left (11 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 16 \,{\left (3 \, b^{3} c + 4 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (8 \,{\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{4} x^{5} + 20 \,{\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{4} x^{4} + 2 \,{\left (9 \, b^{8} c^{2} - 104 \, a b^{6} c^{3} + 384 \, a^{2} b^{4} c^{4} - 384 \, a^{3} b^{2} c^{5} - 256 \, a^{4} c^{6}\right )} d^{4} x^{3} +{\left (7 \, b^{9} c - 72 \, a b^{7} c^{2} + 192 \, a^{2} b^{5} c^{3} + 128 \, a^{3} b^{3} c^{4} - 768 \, a^{4} b c^{5}\right )} d^{4} x^{2} +{\left (b^{10} - 6 \, a b^{8} c - 24 \, a^{2} b^{6} c^{2} + 224 \, a^{3} b^{4} c^{3} - 384 \, a^{4} b^{2} c^{4}\right )} d^{4} x +{\left (a b^{9} - 12 \, a^{2} b^{7} c + 48 \, a^{3} b^{5} c^{2} - 64 \, a^{4} b^{3} c^{3}\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)^(3/2),x, algorithm="fricas")

[Out]

-2/3*(128*c^4*x^4 + 256*b*c^3*x^3 + 3*b^4 + 24*a*b^2*c - 16*a^2*c^2 + 16*(11*b^2*c^2 + 4*a*c^3)*x^2 + 16*(3*b^
3*c + 4*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a)/(8*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^4*x^5 +
20*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^4*x^4 + 2*(9*b^8*c^2 - 104*a*b^6*c^3 + 384*a^2*b
^4*c^4 - 384*a^3*b^2*c^5 - 256*a^4*c^6)*d^4*x^3 + (7*b^9*c - 72*a*b^7*c^2 + 192*a^2*b^5*c^3 + 128*a^3*b^3*c^4
- 768*a^4*b*c^5)*d^4*x^2 + (b^10 - 6*a*b^8*c - 24*a^2*b^6*c^2 + 224*a^3*b^4*c^3 - 384*a^4*b^2*c^4)*d^4*x + (a*
b^9 - 12*a^2*b^7*c + 48*a^3*b^5*c^2 - 64*a^4*b^3*c^3)*d^4)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a b^{4} \sqrt{a + b x + c x^{2}} + 8 a b^{3} c x \sqrt{a + b x + c x^{2}} + 24 a b^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 32 a b c^{3} x^{3} \sqrt{a + b x + c x^{2}} + 16 a c^{4} x^{4} \sqrt{a + b x + c x^{2}} + b^{5} x \sqrt{a + b x + c x^{2}} + 9 b^{4} c x^{2} \sqrt{a + b x + c x^{2}} + 32 b^{3} c^{2} x^{3} \sqrt{a + b x + c x^{2}} + 56 b^{2} c^{3} x^{4} \sqrt{a + b x + c x^{2}} + 48 b c^{4} x^{5} \sqrt{a + b x + c x^{2}} + 16 c^{5} x^{6} \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)**(3/2),x)

[Out]

Integral(1/(a*b**4*sqrt(a + b*x + c*x**2) + 8*a*b**3*c*x*sqrt(a + b*x + c*x**2) + 24*a*b**2*c**2*x**2*sqrt(a +
b*x + c*x**2) + 32*a*b*c**3*x**3*sqrt(a + b*x + c*x**2) + 16*a*c**4*x**4*sqrt(a + b*x + c*x**2) + b**5*x*sqrt
(a + b*x + c*x**2) + 9*b**4*c*x**2*sqrt(a + b*x + c*x**2) + 32*b**3*c**2*x**3*sqrt(a + b*x + c*x**2) + 56*b**2
*c**3*x**4*sqrt(a + b*x + c*x**2) + 48*b*c**4*x**5*sqrt(a + b*x + c*x**2) + 16*c**5*x**6*sqrt(a + b*x + c*x**2
)), x)/d**4

________________________________________________________________________________________

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)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError