3.1281 \(\int \frac{1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=129 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4}}+\frac{4}{d \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}} \]

[Out]

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

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Rubi [A]  time = 0.283319, antiderivative size = 129, 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 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4}}+\frac{4}{d \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 63.7889, size = 126, normalized size = 0.98 \[ \frac{4}{d \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x}} + \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{3}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{3}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.116265, size = 115, normalized size = 0.89 \[ \frac{2 \left (\sqrt{b+2 c x} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-\sqrt{b+2 c x} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+2 \sqrt [4]{b^2-4 a c}\right )}{d \left (b^2-4 a c\right )^{5/4} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^(3/2)/(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)^(1/4)*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)^(1/4)*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)^(1/4)
*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/d/(4
*a*c-b^2)/(2*c*d*x+b*d)^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240842, size = 992, normalized size = 7.69 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(4*sqrt(2*c*d*x + b*d)*(b^2 - 4*a*c)*d*(1/((b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2
- 640*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*d^6))^(1/4)*arctan((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^5*(1/((b^10 - 20*
a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*d
^6))^(3/4)/(sqrt((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^4*sqrt(1/((b
^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a
^5*c^5)*d^6)) + 2*c*d*x + b*d) + sqrt(2*c*d*x + b*d))) + sqrt(2*c*d*x + b*d)*(b^
2 - 4*a*c)*d*(1/((b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*a
^4*b^2*c^4 - 1024*a^5*c^5)*d^6))^(1/4)*log((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^5*(1/((b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 64
0*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*d^6))^(3/4) + sqrt(2*c*d*x + b*
d)) - sqrt(2*c*d*x + b*d)*(b^2 - 4*a*c)*d*(1/((b^10 - 20*a*b^8*c + 160*a^2*b^6*c
^2 - 640*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*d^6))^(1/4)*log(-(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^5*(1/((b^10 - 20*
a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*d
^6))^(3/4) + sqrt(2*c*d*x + b*d)) - 4)/(sqrt(2*c*d*x + b*d)*(b^2 - 4*a*c)*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d \left (b + 2 c x\right )\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((d*(b + 2*c*x))**(3/2)*(a + b*x + c*x**2)), x)

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GIAC/XCAS [A]  time = 0.229762, size = 671, normalized size = 5.2 \[ -\frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}} - \frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}} + \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{4} d^{3} - 8 \, \sqrt{2} a b^{2} c d^{3} + 16 \, \sqrt{2} a^{2} c^{2} d^{3}} - \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{4} d^{3} - 8 \, \sqrt{2} a b^{2} c d^{3} + 16 \, \sqrt{2} a^{2} c^{2} d^{3}} + \frac{4}{{\left (b^{2} d - 4 \, a c d\right )} \sqrt{2 \, c d x + b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*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)^(3/4)*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)^(3/4)*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)^(3/4)*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/
((b^2*d - 4*a*c*d)*sqrt(2*c*d*x + b*d))