3.1238 \(\int \frac{1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

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))

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Rubi [A]  time = 0.172965, 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 \[ -\frac{64 c \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)}-\frac{32 c \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3}-\frac{2}{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt{a+b x+c x^2}} \]

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))

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Rubi in Sympy [A]  time = 42.9192, size = 116, normalized size = 0.98 \[ - \frac{64 c \sqrt{a + b x + c x^{2}}}{3 d^{4} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{3}} - \frac{32 c \sqrt{a + b x + c x^{2}}}{3 d^{4} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )^{2}} - \frac{2}{d^{4} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-64*c*sqrt(a + b*x + c*x**2)/(3*d**4*(b + 2*c*x)*(-4*a*c + b**2)**3) - 32*c*sqrt
(a + b*x + c*x**2)/(3*d**4*(b + 2*c*x)**3*(-4*a*c + b**2)**2) - 2/(d**4*(b + 2*c
*x)**3*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2))

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Mathematica [A]  time = 0.215811, size = 106, normalized size = 0.9 \[ \frac{\left (a+b x+c x^2\right )^2 \left (-\frac{2 (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{40 c}{3 \left (b^2-4 a c\right )^3 (b+2 c x)}-\frac{8 c}{3 \left (b^2-4 a c\right )^2 (b+2 c x)^3}\right )}{d^4 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x + c*x^2)^2*((-8*c)/(3*(b^2 - 4*a*c)^2*(b + 2*c*x)^3) - (40*c)/(3*(b^2
- 4*a*c)^3*(b + 2*c*x)) - (2*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + b*x + c*x^2))))/
(d^4*(a + x*(b + c*x))^(3/2))

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Maple [A]  time = 0.014, size = 133, normalized size = 1.1 \[ -{\frac{-256\,{c}^{4}{x}^{4}-512\,b{c}^{3}{x}^{3}-128\,a{c}^{3}{x}^{2}-352\,{b}^{2}{c}^{2}{x}^{2}-128\,ab{c}^{2}x-96\,{b}^{3}cx+32\,{a}^{2}{c}^{2}-48\,ac{b}^{2}-6\,{b}^{4}}{3\, \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){d}^{4} \left ( 2\,cx+b \right ) ^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \]

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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

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Fricas [A]  time = 0.594534, size = 514, normalized size = 4.36 \[ -\frac{2 \,{\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} + 3 \, b^{4} + 24 \, a b^{2} c - 16 \, a^{2} c^{2} + 16 \,{\left (11 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 16 \,{\left (3 \, b^{3} c + 4 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (8 \,{\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{4} x^{5} + 20 \,{\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{4} x^{4} + 2 \,{\left (9 \, b^{8} c^{2} - 104 \, a b^{6} c^{3} + 384 \, a^{2} b^{4} c^{4} - 384 \, a^{3} b^{2} c^{5} - 256 \, a^{4} c^{6}\right )} d^{4} x^{3} +{\left (7 \, b^{9} c - 72 \, a b^{7} c^{2} + 192 \, a^{2} b^{5} c^{3} + 128 \, a^{3} b^{3} c^{4} - 768 \, a^{4} b c^{5}\right )} d^{4} x^{2} +{\left (b^{10} - 6 \, a b^{8} c - 24 \, a^{2} b^{6} c^{2} + 224 \, a^{3} b^{4} c^{3} - 384 \, a^{4} b^{2} c^{4}\right )} d^{4} x +{\left (a b^{9} - 12 \, a^{2} b^{7} c + 48 \, a^{3} b^{5} c^{2} - 64 \, a^{4} b^{3} c^{3}\right )} d^{4}\right )}} \]

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)

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

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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