3.1289 \(\int \frac{(b d+2 c d x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=131 \[ \frac{6 c d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac{6 c d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac{d (b d+2 c d x)^{3/2}}{a+b x+c x^2} \]

[Out]

-((d*(b*d + 2*c*d*x)^(3/2))/(a + b*x + c*x^2)) + (6*c*d^(5/2)*ArcTan[Sqrt[d*(b +
2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(1/4) - (6*c*d^(5/2)*ArcT
anh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(1/4)

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Rubi [A]  time = 0.26366, 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 \[ \frac{6 c d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac{6 c d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac{d (b d+2 c d x)^{3/2}}{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

-((d*(b*d + 2*c*d*x)^(3/2))/(a + b*x + c*x^2)) + (6*c*d^(5/2)*ArcTan[Sqrt[d*(b +
2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(1/4) - (6*c*d^(5/2)*ArcT
anh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(1/4)

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Rubi in Sympy [A]  time = 63.617, size = 129, normalized size = 0.98 \[ \frac{6 c d^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\sqrt [4]{- 4 a c + b^{2}}} - \frac{6 c d^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\sqrt [4]{- 4 a c + b^{2}}} - \frac{d \left (b d + 2 c d x\right )^{\frac{3}{2}}}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*c*d**(5/2)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(-4*a*c
+ b**2)**(1/4) - 6*c*d**(5/2)*atanh(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)
**(1/4)))/(-4*a*c + b**2)**(1/4) - d*(b*d + 2*c*d*x)**(3/2)/(a + b*x + c*x**2)

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Mathematica [A]  time = 0.354039, size = 143, normalized size = 1.09 \[ \frac{(d (b+2 c x))^{5/2} \left (-\sqrt [4]{b^2-4 a c} (b+2 c x)^{3/2}+6 c (a+x (b+c x)) \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-6 c (a+x (b+c x)) \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )}{\sqrt [4]{b^2-4 a c} (b+2 c x)^{5/2} (a+x (b+c x))} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(5/2)*(-((b^2 - 4*a*c)^(1/4)*(b + 2*c*x)^(3/2)) + 6*c*(a + x*(b
+ c*x))*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] - 6*c*(a + x*(b + c*x))*Arc
Tanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/((b^2 - 4*a*c)^(1/4)*(b + 2*c*x)^(5/
2)*(a + x*(b + c*x)))

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Maple [B]  time = 0.017, size = 327, normalized size = 2.5 \[ -4\,{\frac{c{d}^{3} \left ( 2\,cdx+bd \right ) ^{3/2}}{4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2}}}+{\frac{3\,c{d}^{3}\sqrt{2}}{2}\ln \left ({1 \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+3\,{\frac{c{d}^{3}\sqrt{2}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\arctan \left ({\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }-3\,{\frac{c{d}^{3}\sqrt{2}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\arctan \left ( -{\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232808, size = 460, normalized size = 3.51 \[ \frac{12 \, \left (\frac{c^{4} d^{10}}{b^{2} - 4 \, a c}\right )^{\frac{1}{4}}{\left (c x^{2} + b x + a\right )} \arctan \left (-\frac{\left (\frac{c^{4} d^{10}}{b^{2} - 4 \, a c}\right )^{\frac{3}{4}}{\left (b^{2} - 4 \, a c\right )}}{\sqrt{2 \, c d x + b d} c^{3} d^{7} + \sqrt{2 \, c^{7} d^{15} x + b c^{6} d^{15} + \sqrt{\frac{c^{4} d^{10}}{b^{2} - 4 \, a c}}{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{10}}}\right ) - 3 \, \left (\frac{c^{4} d^{10}}{b^{2} - 4 \, a c}\right )^{\frac{1}{4}}{\left (c x^{2} + b x + a\right )} \log \left (27 \, \sqrt{2 \, c d x + b d} c^{3} d^{7} + 27 \, \left (\frac{c^{4} d^{10}}{b^{2} - 4 \, a c}\right )^{\frac{3}{4}}{\left (b^{2} - 4 \, a c\right )}\right ) + 3 \, \left (\frac{c^{4} d^{10}}{b^{2} - 4 \, a c}\right )^{\frac{1}{4}}{\left (c x^{2} + b x + a\right )} \log \left (27 \, \sqrt{2 \, c d x + b d} c^{3} d^{7} - 27 \, \left (\frac{c^{4} d^{10}}{b^{2} - 4 \, a c}\right )^{\frac{3}{4}}{\left (b^{2} - 4 \, a c\right )}\right ) -{\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt{2 \, c d x + b d}}{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(12*(c^4*d^10/(b^2 - 4*a*c))^(1/4)*(c*x^2 + b*x + a)*arctan(-(c^4*d^10/(b^2 - 4*
a*c))^(3/4)*(b^2 - 4*a*c)/(sqrt(2*c*d*x + b*d)*c^3*d^7 + sqrt(2*c^7*d^15*x + b*c
^6*d^15 + sqrt(c^4*d^10/(b^2 - 4*a*c))*(b^2*c^4 - 4*a*c^5)*d^10))) - 3*(c^4*d^10
/(b^2 - 4*a*c))^(1/4)*(c*x^2 + b*x + a)*log(27*sqrt(2*c*d*x + b*d)*c^3*d^7 + 27*
(c^4*d^10/(b^2 - 4*a*c))^(3/4)*(b^2 - 4*a*c)) + 3*(c^4*d^10/(b^2 - 4*a*c))^(1/4)
*(c*x^2 + b*x + a)*log(27*sqrt(2*c*d*x + b*d)*c^3*d^7 - 27*(c^4*d^10/(b^2 - 4*a*
c))^(3/4)*(b^2 - 4*a*c)) - (2*c*d^2*x + b*d^2)*sqrt(2*c*d*x + b*d))/(c*x^2 + b*x
+ a)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.243145, size = 593, normalized size = 4.53 \[ \frac{4 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c d^{3}}{b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}} - \frac{3 \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{2} - 4 \, a c} - \frac{3 \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{2} - 4 \, a c} + \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c d{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{2} - 4 \, \sqrt{2} a c} - \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c d{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{2} - 4 \, \sqrt{2} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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