3.1277 \(\int \frac{(b d+2 c d x)^{5/2}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=119 \[ 2 d^{5/2} \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{5/2} \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+\frac{4}{3} d (b d+2 c d x)^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.258435, antiderivative size = 119, 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 \[ 2 d^{5/2} \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{5/2} \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+\frac{4}{3} d (b d+2 c d x)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 60.5982, size = 119, normalized size = 1. \[ 2 d^{\frac{5}{2}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 2 d^{\frac{5}{2}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + \frac{4 d \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.183972, size = 131, normalized size = 1.1 \[ \frac{2 (d (b+2 c x))^{5/2} \left (2 \sqrt [4]{b^2-4 a c} (b+2 c x)^{3/2}+3 \left (b^2-4 a c\right ) \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-3 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )}{3 \sqrt [4]{b^2-4 a c} (b+2 c x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 582, normalized size = 4.9 \[{\frac{4\,d}{3} \left ( 2\,cdx+bd \right ) ^{{\frac{3}{2}}}}+4\,{\frac{{d}^{3}\sqrt{2}ac}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\arctan \left ( -{\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }-{{d}^{3}\sqrt{2}{b}^{2}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}-2\,{\frac{{d}^{3}\sqrt{2}ac}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\ln \left ({\frac{2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}{2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}} \right ) }+{\frac{{d}^{3}\sqrt{2}{b}^{2}}{2}\ln \left ({1 \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}-4\,{\frac{{d}^{3}\sqrt{2}ac}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\arctan \left ({\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }+{{d}^{3}\sqrt{2}{b}^{2}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}} \]

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),x)

[Out]

4/3*d*(2*c*d*x+b*d)^(3/2)+4*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)*a*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)*b^
2-2*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)))*a*c+1/
2*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)))*b^2-4*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)*a*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)*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((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 = 0.23829, size = 717, normalized size = 6.03 \[ \frac{4}{3} \,{\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt{2 \, c d x + b d} - 4 \, \left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac{1}{4}} \arctan \left (\frac{\left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac{3}{4}}}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{2 \, c d x + b d} d^{7} + \sqrt{2 \,{\left (b^{8} c - 16 \, a b^{6} c^{2} + 96 \, a^{2} b^{4} c^{3} - 256 \, a^{3} b^{2} c^{4} + 256 \, a^{4} c^{5}\right )} d^{15} x +{\left (b^{9} - 16 \, a b^{7} c + 96 \, a^{2} b^{5} c^{2} - 256 \, a^{3} b^{3} c^{3} + 256 \, a^{4} b c^{4}\right )} d^{15} + \sqrt{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}}{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}}}\right ) - \left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac{1}{4}} \log \left ({\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{2 \, c d x + b d} d^{7} + \left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac{3}{4}}\right ) + \left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac{1}{4}} \log \left ({\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{2 \, c d x + b d} d^{7} - \left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac{3}{4}}\right ) \]

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),x, algorithm="fricas")

[Out]

4/3*(2*c*d^2*x + b*d^2)*sqrt(2*c*d*x + b*d) - 4*((b^6 - 12*a*b^4*c + 48*a^2*b^2*
c^2 - 64*a^3*c^3)*d^10)^(1/4)*arctan(((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^
3*c^3)*d^10)^(3/4)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(2*c*d*x + b*d)*d^7 + sqr
t(2*(b^8*c - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^2*c^4 + 256*a^4*c^5)*d^15
*x + (b^9 - 16*a*b^7*c + 96*a^2*b^5*c^2 - 256*a^3*b^3*c^3 + 256*a^4*b*c^4)*d^15
+ sqrt((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^10)*(b^6 - 12*a*b^4*c
+ 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^10))) - ((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 6
4*a^3*c^3)*d^10)^(1/4)*log((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(2*c*d*x + b*d)*d^
7 + ((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^10)^(3/4)) + ((b^6 - 12*
a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^10)^(1/4)*log((b^4 - 8*a*b^2*c + 16*a^2
*c^2)*sqrt(2*c*d*x + b*d)*d^7 - ((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3
)*d^10)^(3/4))

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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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.252567, size = 478, normalized size = 4.02 \[ -\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) + \frac{1}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} d{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - \frac{1}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} d{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac{4}{3} \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} d \]

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),x, algorithm="giac")

[Out]

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