3.1202 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx\)

Optimal. Leaf size=165 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{256 c^{5/2} d^7 \left (b^2-4 a c\right )^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^7 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{\sqrt{a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(64*c^2*d^7*(b + 2*c*x)^4) + Sqrt[a + b*x + c*x^2]/(128*c
^2*(b^2 - 4*a*c)*d^7*(b + 2*c*x)^2) - (a + b*x + c*x^2)^(3/2)/(12*c*d^7*(b + 2*c
*x)^6) + ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]]/(256*c^(5/2
)*(b^2 - 4*a*c)^(3/2)*d^7)

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Rubi [A]  time = 0.299492, antiderivative size = 165, 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 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{256 c^{5/2} d^7 \left (b^2-4 a c\right )^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^7 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{\sqrt{a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^7,x]

[Out]

-Sqrt[a + b*x + c*x^2]/(64*c^2*d^7*(b + 2*c*x)^4) + Sqrt[a + b*x + c*x^2]/(128*c
^2*(b^2 - 4*a*c)*d^7*(b + 2*c*x)^2) - (a + b*x + c*x^2)^(3/2)/(12*c*d^7*(b + 2*c
*x)^6) + ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]]/(256*c^(5/2
)*(b^2 - 4*a*c)^(3/2)*d^7)

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Rubi in Sympy [A]  time = 75.1057, size = 153, normalized size = 0.93 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{12 c d^{7} \left (b + 2 c x\right )^{6}} + \frac{\sqrt{a + b x + c x^{2}}}{128 c^{2} d^{7} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )} - \frac{\sqrt{a + b x + c x^{2}}}{64 c^{2} d^{7} \left (b + 2 c x\right )^{4}} + \frac{\operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{256 c^{\frac{5}{2}} d^{7} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**7,x)

[Out]

-(a + b*x + c*x**2)**(3/2)/(12*c*d**7*(b + 2*c*x)**6) + sqrt(a + b*x + c*x**2)/(
128*c**2*d**7*(b + 2*c*x)**2*(-4*a*c + b**2)) - sqrt(a + b*x + c*x**2)/(64*c**2*
d**7*(b + 2*c*x)**4) + atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2)
)/(256*c**(5/2)*d**7*(-4*a*c + b**2)**(3/2))

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Mathematica [A]  time = 0.918345, size = 210, normalized size = 1.27 \[ \frac{\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 c^2 \left (8 a^2+14 a c x^2+3 c^2 x^4\right )-8 b^2 c \left (a-2 c x^2\right )+32 b c^2 x \left (7 a+3 c x^2\right )-3 b^4-32 b^3 c x\right )}{\left (b^2-4 a c\right ) (b+2 c x)^6}+\frac{3 \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )}{\left (4 a c-b^2\right )^{3/2}}-\frac{3 \log (b+2 c x)}{\left (4 a c-b^2\right )^{3/2}}}{768 c^{5/2} d^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^7,x]

[Out]

((2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3*b^4 - 32*b^3*c*x - 8*b^2*c*(a - 2*c*x^2) +
32*b*c^2*x*(7*a + 3*c*x^2) + 16*c^2*(8*a^2 + 14*a*c*x^2 + 3*c^2*x^4)))/((b^2 -
4*a*c)*(b + 2*c*x)^6) - (3*Log[b + 2*c*x])/(-b^2 + 4*a*c)^(3/2) + (3*Log[-(b^2*S
qrt[c]) + 4*a*c^(3/2) + 2*c*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]])/(-b^2 + 4
*a*c)^(3/2))/(768*c^(5/2)*d^7)

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Maple [B]  time = 0.031, size = 682, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^7,x)

[Out]

-1/192/d^7/c^6/(4*a*c-b^2)/(x+1/2*b/c)^6*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/
2)+1/192/d^7/c^4/(4*a*c-b^2)^2/(x+1/2*b/c)^4*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)
^(5/2)+1/96/d^7/c^2/(4*a*c-b^2)^3/(x+1/2*b/c)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)
/c)^(5/2)-1/96/d^7/c/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)-1/6
4/d^7/c/(4*a*c-b^2)^3*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*a+1/256/d^7/c^2/(4
*a*c-b^2)^3*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*b^2+1/16/d^7/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))*a^2-1/32/d^7/c^2/(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))*a*b^2+1/256/d^7/c^3/(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))*b^4

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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((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.24132, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^7,x, algorithm="fricas")

[Out]

[1/1536*(4*(48*c^4*x^4 + 96*b*c^3*x^3 - 3*b^4 - 8*a*b^2*c + 128*a^2*c^2 + 16*(b^
2*c^2 + 14*a*c^3)*x^2 - 32*(b^3*c - 7*a*b*c^2)*x)*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*
x^2 + b*x + a) - 3*(64*c^6*x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x
^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^6)*log(-((4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c
)*sqrt(-b^2*c + 4*a*c^2) - 4*(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a))/(4*c^2*x^2
+ 4*b*c*x + b^2)))/((64*(b^2*c^8 - 4*a*c^9)*d^7*x^6 + 192*(b^3*c^7 - 4*a*b*c^8)
*d^7*x^5 + 240*(b^4*c^6 - 4*a*b^2*c^7)*d^7*x^4 + 160*(b^5*c^5 - 4*a*b^3*c^6)*d^7
*x^3 + 60*(b^6*c^4 - 4*a*b^4*c^5)*d^7*x^2 + 12*(b^7*c^3 - 4*a*b^5*c^4)*d^7*x + (
b^8*c^2 - 4*a*b^6*c^3)*d^7)*sqrt(-b^2*c + 4*a*c^2)), 1/768*(2*(48*c^4*x^4 + 96*b
*c^3*x^3 - 3*b^4 - 8*a*b^2*c + 128*a^2*c^2 + 16*(b^2*c^2 + 14*a*c^3)*x^2 - 32*(b
^3*c - 7*a*b*c^2)*x)*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a) - 3*(64*c^6*x^6
+ 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c
*x + b^6)*arctan(1/2*sqrt(b^2*c - 4*a*c^2)/(sqrt(c*x^2 + b*x + a)*c)))/((64*(b^2
*c^8 - 4*a*c^9)*d^7*x^6 + 192*(b^3*c^7 - 4*a*b*c^8)*d^7*x^5 + 240*(b^4*c^6 - 4*a
*b^2*c^7)*d^7*x^4 + 160*(b^5*c^5 - 4*a*b^3*c^6)*d^7*x^3 + 60*(b^6*c^4 - 4*a*b^4*
c^5)*d^7*x^2 + 12*(b^7*c^3 - 4*a*b^5*c^4)*d^7*x + (b^8*c^2 - 4*a*b^6*c^3)*d^7)*s
qrt(b^2*c - 4*a*c^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{a \sqrt{a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac{b x \sqrt{a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac{c x^{2} \sqrt{a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx}{d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**7,x)

[Out]

(Integral(a*sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 + 280
*b**4*c**3*x**3 + 560*b**3*c**4*x**4 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 12
8*c**7*x**7), x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x + 84*
b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*c**4*x**4 + 672*b**2*c**5*x**5 +
448*b*c**6*x**6 + 128*c**7*x**7), x) + Integral(c*x**2*sqrt(a + b*x + c*x**2)/(b
**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*c**4*x**4
+ 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 128*c**7*x**7), x))/d**7

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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((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^7,x, algorithm="giac")

[Out]

Exception raised: TypeError