3.1190 \(\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^7} \, dx\)

Optimal. Leaf size=175 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{64 c^{3/2} d^7 \left (b^2-4 a c\right )^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{32 c d^7 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac{\sqrt{a+b x+c x^2}}{48 c d^7 \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac{\sqrt{a+b x+c x^2}}{12 c d^7 (b+2 c x)^6} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(12*c*d^7*(b + 2*c*x)^6) + Sqrt[a + b*x + c*x^2]/(48*c*(b
^2 - 4*a*c)*d^7*(b + 2*c*x)^4) + Sqrt[a + b*x + c*x^2]/(32*c*(b^2 - 4*a*c)^2*d^7
*(b + 2*c*x)^2) + ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]]/(6
4*c^(3/2)*(b^2 - 4*a*c)^(5/2)*d^7)

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Rubi [A]  time = 0.314436, antiderivative size = 175, 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 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{64 c^{3/2} d^7 \left (b^2-4 a c\right )^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{32 c d^7 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac{\sqrt{a+b x+c x^2}}{48 c d^7 \left (b^2-4 a c\right ) (b+2 c x)^4}-\frac{\sqrt{a+b x+c x^2}}{12 c d^7 (b+2 c x)^6} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + b*x + c*x^2]/(12*c*d^7*(b + 2*c*x)^6) + Sqrt[a + b*x + c*x^2]/(48*c*(b
^2 - 4*a*c)*d^7*(b + 2*c*x)^4) + Sqrt[a + b*x + c*x^2]/(32*c*(b^2 - 4*a*c)^2*d^7
*(b + 2*c*x)^2) + ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]]/(6
4*c^(3/2)*(b^2 - 4*a*c)^(5/2)*d^7)

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Rubi in Sympy [A]  time = 75.5175, size = 160, normalized size = 0.91 \[ \frac{\sqrt{a + b x + c x^{2}}}{32 c d^{7} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )^{2}} + \frac{\sqrt{a + b x + c x^{2}}}{48 c d^{7} \left (b + 2 c x\right )^{4} \left (- 4 a c + b^{2}\right )} - \frac{\sqrt{a + b x + c x^{2}}}{12 c d^{7} \left (b + 2 c x\right )^{6}} + \frac{\operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{64 c^{\frac{3}{2}} d^{7} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**7,x)

[Out]

sqrt(a + b*x + c*x**2)/(32*c*d**7*(b + 2*c*x)**2*(-4*a*c + b**2)**2) + sqrt(a +
b*x + c*x**2)/(48*c*d**7*(b + 2*c*x)**4*(-4*a*c + b**2)) - sqrt(a + b*x + c*x**2
)/(12*c*d**7*(b + 2*c*x)**6) + atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c
+ b**2))/(64*c**(3/2)*d**7*(-4*a*c + b**2)**(5/2))

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Mathematica [A]  time = 0.954926, size = 172, normalized size = 0.98 \[ \frac{-\frac{3 \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (\frac{3 (b+2 c x)^4}{\left (b^2-4 a c\right )^2}+\frac{2 (b+2 c x)^2}{b^2-4 a c}-8\right )}{(b+2 c x)^6}+\frac{3 \log (b+2 c x)}{\left (4 a c-b^2\right )^{5/2}}}{192 c^{3/2} d^7} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-8 + (2*(b + 2*c*x)^2)/(b^2 - 4*a*c) + (3*(b
+ 2*c*x)^4)/(b^2 - 4*a*c)^2))/(b + 2*c*x)^6 + (3*Log[b + 2*c*x])/(-b^2 + 4*a*c)^
(5/2) - (3*Log[-(b^2*Sqrt[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)^(5/2))/(192*c^(3/2)*d^7)

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Maple [B]  time = 0.034, size = 460, normalized size = 2.6 \[ -{\frac{1}{192\,{d}^{7}{c}^{6} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-6}}+{\frac{1}{64\,{c}^{4}{d}^{7} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-4}}-{\frac{1}{32\,{d}^{7}{c}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}}+{\frac{1}{64\,{d}^{7}c \left ( 4\,ac-{b}^{2} \right ) ^{3}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{a}{16\,{d}^{7}c \left ( 4\,ac-{b}^{2} \right ) ^{3}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{{b}^{2}}{64\,{d}^{7}{c}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2)/(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)^(3/
2)+1/64/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)^
(3/2)-1/32/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)^(3/2)+1/64/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)-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+1/64/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))*b^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(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2734, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(4*(48*c^4*x^4 + 96*b*c^3*x^3 - 3*b^4 + 56*a*b^2*c - 128*a^2*c^2 + 16*(5*
b^2*c^2 - 2*a*c^3)*x^2 + 32*(b^3*c - 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^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)*d^7*x^6 + 192*(b^5*
c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^8)*d^7*x^5 + 240*(b^6*c^5 - 8*a*b^4*c^6 + 16*a^2*
b^2*c^7)*d^7*x^4 + 160*(b^7*c^4 - 8*a*b^5*c^5 + 16*a^2*b^3*c^6)*d^7*x^3 + 60*(b^
8*c^3 - 8*a*b^6*c^4 + 16*a^2*b^4*c^5)*d^7*x^2 + 12*(b^9*c^2 - 8*a*b^7*c^3 + 16*a
^2*b^5*c^4)*d^7*x + (b^10*c - 8*a*b^8*c^2 + 16*a^2*b^6*c^3)*d^7)*sqrt(-b^2*c + 4
*a*c^2)), 1/192*(2*(48*c^4*x^4 + 96*b*c^3*x^3 - 3*b^4 + 56*a*b^2*c - 128*a^2*c^2
+ 16*(5*b^2*c^2 - 2*a*c^3)*x^2 + 32*(b^3*c - 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^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)*d^7*x^6 +
192*(b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^8)*d^7*x^5 + 240*(b^6*c^5 - 8*a*b^4*c^6
+ 16*a^2*b^2*c^7)*d^7*x^4 + 160*(b^7*c^4 - 8*a*b^5*c^5 + 16*a^2*b^3*c^6)*d^7*x^3
+ 60*(b^8*c^3 - 8*a*b^6*c^4 + 16*a^2*b^4*c^5)*d^7*x^2 + 12*(b^9*c^2 - 8*a*b^7*c
^3 + 16*a^2*b^5*c^4)*d^7*x + (b^10*c - 8*a*b^8*c^2 + 16*a^2*b^6*c^3)*d^7)*sqrt(b
^2*c - 4*a*c^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**7,x)

[Out]

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

[Out]

Exception raised: TypeError