3.1247 \(\int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=176 \[ \frac{40 c^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{7/2}}+\frac{80 c^2 \sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}+\frac{40 c}{3 d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{2}{3 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

-2/(3*(b^2 - 4*a*c)*d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^(3/2)) + (40*c)/(3*(b^2
- 4*a*c)^2*d^3*(b + 2*c*x)^2*Sqrt[a + b*x + c*x^2]) + (80*c^2*Sqrt[a + b*x + c*x
^2])/((b^2 - 4*a*c)^3*d^3*(b + 2*c*x)^2) + (40*c^(3/2)*ArcTan[(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(7/2)*d^3)

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Rubi [A]  time = 0.32376, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{40 c^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{7/2}}+\frac{80 c^2 \sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}+\frac{40 c}{3 d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{2}{3 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(5/2)),x]

[Out]

-2/(3*(b^2 - 4*a*c)*d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^(3/2)) + (40*c)/(3*(b^2
- 4*a*c)^2*d^3*(b + 2*c*x)^2*Sqrt[a + b*x + c*x^2]) + (80*c^2*Sqrt[a + b*x + c*x
^2])/((b^2 - 4*a*c)^3*d^3*(b + 2*c*x)^2) + (40*c^(3/2)*ArcTan[(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(7/2)*d^3)

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Rubi in Sympy [A]  time = 74.3136, size = 173, normalized size = 0.98 \[ \frac{40 c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{d^{3} \left (- 4 a c + b^{2}\right )^{\frac{7}{2}}} + \frac{80 c^{2} \sqrt{a + b x + c x^{2}}}{d^{3} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )^{3}} + \frac{40 c}{3 d^{3} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} - \frac{2}{3 d^{3} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**3/(c*x**2+b*x+a)**(5/2),x)

[Out]

40*c**(3/2)*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2))/(d**3*(-4
*a*c + b**2)**(7/2)) + 80*c**2*sqrt(a + b*x + c*x**2)/(d**3*(b + 2*c*x)**2*(-4*a
*c + b**2)**3) + 40*c/(3*d**3*(b + 2*c*x)**2*(-4*a*c + b**2)**2*sqrt(a + b*x + c
*x**2)) - 2/(3*d**3*(b + 2*c*x)**2*(-4*a*c + b**2)*(a + b*x + c*x**2)**(3/2))

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Mathematica [A]  time = 1.05443, size = 180, normalized size = 1.02 \[ \frac{-\frac{40 c^{3/2} \log (b+2 c x)}{\left (4 a c-b^2\right )^{7/2}}+\frac{40 c^{3/2} \log \left (\sqrt{c} \left (-2 \sqrt{c} \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}-4 a c+b^2\right )\right )}{\left (4 a c-b^2\right )^{7/2}}+\frac{2 \sqrt{a+x (b+c x)} \left (\frac{4 a c-b^2}{(a+x (b+c x))^2}+\frac{24 c}{a+x (b+c x)}+\frac{24 c^2}{(b+2 c x)^2}\right )}{3 \left (b^2-4 a c\right )^3}}{d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(5/2)),x]

[Out]

((2*Sqrt[a + x*(b + c*x)]*((24*c^2)/(b + 2*c*x)^2 + (-b^2 + 4*a*c)/(a + x*(b + c
*x))^2 + (24*c)/(a + x*(b + c*x))))/(3*(b^2 - 4*a*c)^3) - (40*c^(3/2)*Log[b + 2*
c*x])/(-b^2 + 4*a*c)^(7/2) + (40*c^(3/2)*Log[Sqrt[c]*(b^2 - 4*a*c - 2*Sqrt[c]*Sq
rt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)])])/(-b^2 + 4*a*c)^(7/2))/d^3

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Maple [A]  time = 0.024, size = 267, normalized size = 1.5 \[ -{\frac{1}{4\,{c}^{2}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) } \left ( x+{\frac{b}{2\,c}} \right ) ^{-2} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{-{\frac{3}{2}}}}-{\frac{5}{3\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{-{\frac{3}{2}}}}-20\,{\frac{c}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3}}{\frac{1}{\sqrt{ \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+1/4\,{\frac{4\,ac-{b}^{2}}{c}}}}}}+40\,{\frac{c}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3}}\ln \left ({1 \left ( 1/2\,{\frac{4\,ac-{b}^{2}}{c}}+1/2\,\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}} \right ) \left ( x+1/2\,{\frac{b}{c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^3/(c*x^2+b*x+a)^(5/2),x)

[Out]

-1/4/d^3/c^2/(4*a*c-b^2)/(x+1/2*b/c)^2/((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)
-5/3/d^3/(4*a*c-b^2)^2/((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)-20/d^3*c/(4*a*c
-b^2)^3/((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)+40/d^3*c/(4*a*c-b^2)^3/((4*a*c
-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*
c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))

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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)^3*(c*x^2 + b*x + a)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.01323, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^(5/2)),x, algorithm="fricas")

[Out]

[-2/3*(30*(4*c^5*x^6 + 12*b*c^4*x^5 + a^2*b^2*c + (13*b^2*c^3 + 8*a*c^4)*x^4 + 2
*(3*b^3*c^2 + 8*a*b*c^3)*x^3 + (b^4*c + 10*a*b^2*c^2 + 4*a^2*c^3)*x^2 + 2*(a*b^3
*c + 2*a^2*b*c^2)*x)*sqrt(-c/(b^2 - 4*a*c))*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*
a*c - 4*sqrt(c*x^2 + b*x + a)*(b^2 - 4*a*c)*sqrt(-c/(b^2 - 4*a*c)))/(4*c^2*x^2 +
4*b*c*x + b^2)) - (120*c^4*x^4 + 240*b*c^3*x^3 - b^4 + 28*a*b^2*c + 24*a^2*c^2
+ 20*(7*b^2*c^2 + 8*a*c^3)*x^2 + 20*(b^3*c + 8*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a)
)/(4*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^3*x^6 + 12*(b^7*c^
3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*x^5 + (13*b^8*c^2 - 148*a*
b^6*c^3 + 528*a^2*b^4*c^4 - 448*a^3*b^2*c^5 - 512*a^4*c^6)*d^3*x^4 + 2*(3*b^9*c
- 28*a*b^7*c^2 + 48*a^2*b^5*c^3 + 192*a^3*b^3*c^4 - 512*a^4*b*c^5)*d^3*x^3 + (b^
10 - 2*a*b^8*c - 68*a^2*b^6*c^2 + 368*a^3*b^4*c^3 - 448*a^4*b^2*c^4 - 256*a^5*c^
5)*d^3*x^2 + 2*(a*b^9 - 10*a^2*b^7*c + 24*a^3*b^5*c^2 + 32*a^4*b^3*c^3 - 128*a^5
*b*c^4)*d^3*x + (a^2*b^8 - 12*a^3*b^6*c + 48*a^4*b^4*c^2 - 64*a^5*b^2*c^3)*d^3),
-2/3*(60*(4*c^5*x^6 + 12*b*c^4*x^5 + a^2*b^2*c + (13*b^2*c^3 + 8*a*c^4)*x^4 + 2
*(3*b^3*c^2 + 8*a*b*c^3)*x^3 + (b^4*c + 10*a*b^2*c^2 + 4*a^2*c^3)*x^2 + 2*(a*b^3
*c + 2*a^2*b*c^2)*x)*sqrt(c/(b^2 - 4*a*c))*arctan(1/2/(sqrt(c*x^2 + b*x + a)*sqr
t(c/(b^2 - 4*a*c)))) - (120*c^4*x^4 + 240*b*c^3*x^3 - b^4 + 28*a*b^2*c + 24*a^2*
c^2 + 20*(7*b^2*c^2 + 8*a*c^3)*x^2 + 20*(b^3*c + 8*a*b*c^2)*x)*sqrt(c*x^2 + b*x
+ a))/(4*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^3*x^6 + 12*(b^
7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*x^5 + (13*b^8*c^2 - 14
8*a*b^6*c^3 + 528*a^2*b^4*c^4 - 448*a^3*b^2*c^5 - 512*a^4*c^6)*d^3*x^4 + 2*(3*b^
9*c - 28*a*b^7*c^2 + 48*a^2*b^5*c^3 + 192*a^3*b^3*c^4 - 512*a^4*b*c^5)*d^3*x^3 +
(b^10 - 2*a*b^8*c - 68*a^2*b^6*c^2 + 368*a^3*b^4*c^3 - 448*a^4*b^2*c^4 - 256*a^
5*c^5)*d^3*x^2 + 2*(a*b^9 - 10*a^2*b^7*c + 24*a^3*b^5*c^2 + 32*a^4*b^3*c^3 - 128
*a^5*b*c^4)*d^3*x + (a^2*b^8 - 12*a^3*b^6*c + 48*a^4*b^4*c^2 - 64*a^5*b^2*c^3)*d
^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{a^{2} b^{3} \sqrt{a + b x + c x^{2}} + 6 a^{2} b^{2} c x \sqrt{a + b x + c x^{2}} + 12 a^{2} b c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 8 a^{2} c^{3} x^{3} \sqrt{a + b x + c x^{2}} + 2 a b^{4} x \sqrt{a + b x + c x^{2}} + 14 a b^{3} c x^{2} \sqrt{a + b x + c x^{2}} + 36 a b^{2} c^{2} x^{3} \sqrt{a + b x + c x^{2}} + 40 a b c^{3} x^{4} \sqrt{a + b x + c x^{2}} + 16 a c^{4} x^{5} \sqrt{a + b x + c x^{2}} + b^{5} x^{2} \sqrt{a + b x + c x^{2}} + 8 b^{4} c x^{3} \sqrt{a + b x + c x^{2}} + 25 b^{3} c^{2} x^{4} \sqrt{a + b x + c x^{2}} + 38 b^{2} c^{3} x^{5} \sqrt{a + b x + c x^{2}} + 28 b c^{4} x^{6} \sqrt{a + b x + c x^{2}} + 8 c^{5} x^{7} \sqrt{a + b x + c x^{2}}}\, dx}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*c*d*x+b*d)**3/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/(a**2*b**3*sqrt(a + b*x + c*x**2) + 6*a**2*b**2*c*x*sqrt(a + b*x + c*
x**2) + 12*a**2*b*c**2*x**2*sqrt(a + b*x + c*x**2) + 8*a**2*c**3*x**3*sqrt(a + b
*x + c*x**2) + 2*a*b**4*x*sqrt(a + b*x + c*x**2) + 14*a*b**3*c*x**2*sqrt(a + b*x
+ c*x**2) + 36*a*b**2*c**2*x**3*sqrt(a + b*x + c*x**2) + 40*a*b*c**3*x**4*sqrt(
a + b*x + c*x**2) + 16*a*c**4*x**5*sqrt(a + b*x + c*x**2) + b**5*x**2*sqrt(a + b
*x + c*x**2) + 8*b**4*c*x**3*sqrt(a + b*x + c*x**2) + 25*b**3*c**2*x**4*sqrt(a +
b*x + c*x**2) + 38*b**2*c**3*x**5*sqrt(a + b*x + c*x**2) + 28*b*c**4*x**6*sqrt(
a + b*x + c*x**2) + 8*c**5*x**7*sqrt(a + b*x + c*x**2)), x)/d**3

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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)^3*(c*x^2 + b*x + a)^(5/2)),x, algorithm="giac")

[Out]

Exception raised: TypeError