3.1217 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^7} \, dx\)

Optimal. Leaf size=155 \[ \frac{5 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{1024 c^{7/2} d^7 \sqrt{b^2-4 a c}}-\frac{5 \sqrt{a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac{\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6} \]

[Out]

(-5*Sqrt[a + b*x + c*x^2])/(512*c^3*d^7*(b + 2*c*x)^2) - (5*(a + b*x + c*x^2)^(3
/2))/(192*c^2*d^7*(b + 2*c*x)^4) - (a + b*x + c*x^2)^(5/2)/(12*c*d^7*(b + 2*c*x)
^6) + (5*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(1024*c^(7
/2)*Sqrt[b^2 - 4*a*c]*d^7)

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Rubi [A]  time = 0.292539, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{5 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{1024 c^{7/2} d^7 \sqrt{b^2-4 a c}}-\frac{5 \sqrt{a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac{\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^7,x]

[Out]

(-5*Sqrt[a + b*x + c*x^2])/(512*c^3*d^7*(b + 2*c*x)^2) - (5*(a + b*x + c*x^2)^(3
/2))/(192*c^2*d^7*(b + 2*c*x)^4) - (a + b*x + c*x^2)^(5/2)/(12*c*d^7*(b + 2*c*x)
^6) + (5*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(1024*c^(7
/2)*Sqrt[b^2 - 4*a*c]*d^7)

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Rubi in Sympy [A]  time = 71.9196, size = 150, normalized size = 0.97 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{12 c d^{7} \left (b + 2 c x\right )^{6}} - \frac{5 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{192 c^{2} d^{7} \left (b + 2 c x\right )^{4}} - \frac{5 \sqrt{a + b x + c x^{2}}}{512 c^{3} d^{7} \left (b + 2 c x\right )^{2}} + \frac{5 \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{1024 c^{\frac{7}{2}} d^{7} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**7,x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(12*c*d**7*(b + 2*c*x)**6) - 5*(a + b*x + c*x**2)**(3
/2)/(192*c**2*d**7*(b + 2*c*x)**4) - 5*sqrt(a + b*x + c*x**2)/(512*c**3*d**7*(b
+ 2*c*x)**2) + 5*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2))/(102
4*c**(7/2)*d**7*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.487821, size = 171, normalized size = 1.1 \[ \frac{-\frac{15 \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )}{\sqrt{4 a c-b^2}}-\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-26 \left (b^2-4 a c\right ) (b+2 c x)^2+8 \left (b^2-4 a c\right )^2+33 (b+2 c x)^4\right )}{(b+2 c x)^6}+\frac{15 \log (b+2 c x)}{\sqrt{4 a c-b^2}}}{3072 c^{7/2} d^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^7,x]

[Out]

((-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(8*(b^2 - 4*a*c)^2 - 26*(b^2 - 4*a*c)*(b + 2*
c*x)^2 + 33*(b + 2*c*x)^4))/(b + 2*c*x)^6 + (15*Log[b + 2*c*x])/Sqrt[-b^2 + 4*a*
c] - (15*Log[-(b^2*Sqrt[c]) + 4*a*c^(3/2) + 2*c*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b
+ c*x)]])/Sqrt[-b^2 + 4*a*c])/(3072*c^(7/2)*d^7)

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Maple [B]  time = 0.033, size = 960, normalized size = 6.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(5/2)/(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)^(7/
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)
^(7/2)-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)^(7/2)+1/32/d^7/c/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+5/9
6/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)*a-5/384/d^7/c^2/
(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*b^2+5/64/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)*a^2-5/128/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)*a*b^2+5/1024/d^7/c^3/(4*a*c-b^2)^3*(4*(x+1/
2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*b^4-5/16/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^3+15/64/d^7/c^2/(4*a*c-b^2)^3/((4*a*c-b^2)/c)^(1/2)*l
n((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*b^2-15/256/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))*a*b^4+5/1024/d^7/c^4/(4*a*c-b^2)^3/((4*a*c-b^2)/c)^(1/2)*l
n((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^6

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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)^(5/2)/(2*c*d*x + b*d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.21981, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (528 \, c^{4} x^{4} + 1056 \, b c^{3} x^{3} + 15 \, b^{4} + 40 \, a b^{2} c + 128 \, a^{2} c^{2} + 16 \,{\left (43 \, b^{2} c^{2} + 26 \, a c^{3}\right )} x^{2} + 32 \,{\left (5 \, b^{3} c + 13 \, a b c^{2}\right )} x\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} \sqrt{c x^{2} + b x + a} - 15 \,{\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \log \left (-\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right )}{6144 \,{\left (64 \, c^{9} d^{7} x^{6} + 192 \, b c^{8} d^{7} x^{5} + 240 \, b^{2} c^{7} d^{7} x^{4} + 160 \, b^{3} c^{6} d^{7} x^{3} + 60 \, b^{4} c^{5} d^{7} x^{2} + 12 \, b^{5} c^{4} d^{7} x + b^{6} c^{3} d^{7}\right )} \sqrt{-b^{2} c + 4 \, a c^{2}}}, -\frac{2 \,{\left (528 \, c^{4} x^{4} + 1056 \, b c^{3} x^{3} + 15 \, b^{4} + 40 \, a b^{2} c + 128 \, a^{2} c^{2} + 16 \,{\left (43 \, b^{2} c^{2} + 26 \, a c^{3}\right )} x^{2} + 32 \,{\left (5 \, b^{3} c + 13 \, a b c^{2}\right )} x\right )} \sqrt{b^{2} c - 4 \, a c^{2}} \sqrt{c x^{2} + b x + a} + 15 \,{\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \arctan \left (\frac{\sqrt{b^{2} c - 4 \, a c^{2}}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{3072 \,{\left (64 \, c^{9} d^{7} x^{6} + 192 \, b c^{8} d^{7} x^{5} + 240 \, b^{2} c^{7} d^{7} x^{4} + 160 \, b^{3} c^{6} d^{7} x^{3} + 60 \, b^{4} c^{5} d^{7} x^{2} + 12 \, b^{5} c^{4} d^{7} x + b^{6} c^{3} d^{7}\right )} \sqrt{b^{2} c - 4 \, a c^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^7,x, algorithm="fricas")

[Out]

[-1/6144*(4*(528*c^4*x^4 + 1056*b*c^3*x^3 + 15*b^4 + 40*a*b^2*c + 128*a^2*c^2 +
16*(43*b^2*c^2 + 26*a*c^3)*x^2 + 32*(5*b^3*c + 13*a*b*c^2)*x)*sqrt(-b^2*c + 4*a*
c^2)*sqrt(c*x^2 + b*x + a) - 15*(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*c^9*d^7*x^6 + 192*b*c^8*d^7*x^5 + 240*b^2*
c^7*d^7*x^4 + 160*b^3*c^6*d^7*x^3 + 60*b^4*c^5*d^7*x^2 + 12*b^5*c^4*d^7*x + b^6*
c^3*d^7)*sqrt(-b^2*c + 4*a*c^2)), -1/3072*(2*(528*c^4*x^4 + 1056*b*c^3*x^3 + 15*
b^4 + 40*a*b^2*c + 128*a^2*c^2 + 16*(43*b^2*c^2 + 26*a*c^3)*x^2 + 32*(5*b^3*c +
13*a*b*c^2)*x)*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a) + 15*(64*c^6*x^6 + 19
2*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*c^9*d^7*x
^6 + 192*b*c^8*d^7*x^5 + 240*b^2*c^7*d^7*x^4 + 160*b^3*c^6*d^7*x^3 + 60*b^4*c^5*
d^7*x^2 + 12*b^5*c^4*d^7*x + b^6*c^3*d^7)*sqrt(b^2*c - 4*a*c^2))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**7,x)

[Out]

Timed out

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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)^(5/2)/(2*c*d*x + b*d)^7,x, algorithm="giac")

[Out]

Exception raised: TypeError