3.1282 \(\int \frac{1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=131 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{7/4}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{7/4}}+\frac{4}{3 d \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}} \]

[Out]

4/(3*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)) - (2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^
2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(7/4)*d^(5/2)) - (2*ArcTanh[Sqrt[d*(b
+ 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(7/4)*d^(5/2))

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Rubi [A]  time = 0.279512, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{7/4}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{7/4}}+\frac{4}{3 d \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

4/(3*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)) - (2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^
2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(7/4)*d^(5/2)) - (2*ArcTanh[Sqrt[d*(b
+ 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(7/4)*d^(5/2))

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Rubi in Sympy [A]  time = 63.7745, size = 128, normalized size = 0.98 \[ \frac{4}{3 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}}} - \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{5}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{5}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/(3*d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(3/2)) - 2*atan(sqrt(b*d + 2*c*d*x)/(sqr
t(d)*(-4*a*c + b**2)**(1/4)))/(d**(5/2)*(-4*a*c + b**2)**(7/4)) - 2*atanh(sqrt(b
*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(d**(5/2)*(-4*a*c + b**2)**(7/4)
)

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Mathematica [A]  time = 0.276224, size = 118, normalized size = 0.9 \[ \frac{2 \left (-3 (b+2 c x)^{3/2} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-3 (b+2 c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+2 \left (b^2-4 a c\right )^{3/4}\right )}{3 d \left (b^2-4 a c\right )^{7/4} (d (b+2 c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(2*(b^2 - 4*a*c)^(3/4) - 3*(b + 2*c*x)^(3/2)*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*
a*c)^(1/4)] - 3*(b + 2*c*x)^(3/2)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))
/(3*(b^2 - 4*a*c)^(7/4)*d*(d*(b + 2*c*x))^(3/2))

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Maple [B]  time = 0.013, size = 341, normalized size = 2.6 \[ -{\frac{\sqrt{2}}{2\,d \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({1 \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}-{\frac{\sqrt{2}}{d \left ( 4\,ac-{b}^{2} \right ) }\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}+{\frac{\sqrt{2}}{d \left ( 4\,ac-{b}^{2} \right ) }\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}-{\frac{4}{3\,d \left ( 4\,ac-{b}^{2} \right ) } \left ( 2\,cdx+bd \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241519, size = 1249, normalized size = 9.53 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(12*(2*(b^2*c - 4*a*c^2)*d^2*x + (b^3 - 4*a*b*c)*d^2)*sqrt(2*c*d*x + b*d)*(1
/((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 -
21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^10))^(1/4)*arctan((b^4
- 8*a*b^2*c + 16*a^2*c^2)*d^3*(1/((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240
*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*
a^7*c^7)*d^10))^(1/4)/(sqrt((b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3
+ 256*a^4*c^4)*d^6*sqrt(1/((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^
8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7
)*d^10)) + 2*c*d*x + b*d) + sqrt(2*c*d*x + b*d))) - 3*(2*(b^2*c - 4*a*c^2)*d^2*x
+ (b^3 - 4*a*b*c)*d^2)*sqrt(2*c*d*x + b*d)*(1/((b^14 - 28*a*b^12*c + 336*a^2*b^
10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2
*c^6 - 16384*a^7*c^7)*d^10))^(1/4)*log((b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^3*(1/((b
^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 215
04*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^10))^(1/4) + sqrt(2*c*d*x
+ b*d)) + 3*(2*(b^2*c - 4*a*c^2)*d^2*x + (b^3 - 4*a*b*c)*d^2)*sqrt(2*c*d*x + b*d
)*(1/((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c
^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^10))^(1/4)*log(-(b
^4 - 8*a*b^2*c + 16*a^2*c^2)*d^3*(1/((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 22
40*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 1638
4*a^7*c^7)*d^10))^(1/4) + sqrt(2*c*d*x + b*d)) + 4)/((2*(b^2*c - 4*a*c^2)*d^2*x
+ (b^3 - 4*a*b*c)*d^2)*sqrt(2*c*d*x + b*d))

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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(1/(2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230991, size = 671, normalized size = 5.12 \[ -\frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}} - \frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}} - \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{4} d^{3} - 8 \, \sqrt{2} a b^{2} c d^{3} + 16 \, \sqrt{2} a^{2} c^{2} d^{3}} + \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{4} d^{3} - 8 \, \sqrt{2} a b^{2} c d^{3} + 16 \, \sqrt{2} a^{2} c^{2} d^{3}} + \frac{4}{3 \,{\left (b^{2} d - 4 \, a c d\right )}{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*
a*c*d^2)^(1/4) + 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4))/(b^4*d^3 -
8*a*b^2*c*d^3 + 16*a^2*c^2*d^3) - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*arctan(-
1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) - 2*sqrt(2*c*d*x + b*d))/(-b^2
*d^2 + 4*a*c*d^2)^(1/4))/(b^4*d^3 - 8*a*b^2*c*d^3 + 16*a^2*c^2*d^3) - (-b^2*d^2
+ 4*a*c*d^2)^(1/4)*ln(2*c*d*x + b*d + sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(
2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2))/(sqrt(2)*b^4*d^3 - 8*sqrt(2)*a*b^2*
c*d^3 + 16*sqrt(2)*a^2*c^2*d^3) + (-b^2*d^2 + 4*a*c*d^2)^(1/4)*ln(2*c*d*x + b*d
- sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a
*c*d^2))/(sqrt(2)*b^4*d^3 - 8*sqrt(2)*a*b^2*c*d^3 + 16*sqrt(2)*a^2*c^2*d^3) + 4/
3/((b^2*d - 4*a*c*d)*(2*c*d*x + b*d)^(3/2))