3.1288 \(\int \frac{(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=150 \[ -10 c d^{7/2} \sqrt [4]{b^2-4 a c} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-10 c d^{7/2} \sqrt [4]{b^2-4 a c} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-\frac{d (b d+2 c d x)^{5/2}}{a+b x+c x^2}+20 c d^3 \sqrt{b d+2 c d x} \]

[Out]

20*c*d^3*Sqrt[b*d + 2*c*d*x] - (d*(b*d + 2*c*d*x)^(5/2))/(a + b*x + c*x^2) - 10*
c*(b^2 - 4*a*c)^(1/4)*d^(7/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sq
rt[d])] - 10*c*(b^2 - 4*a*c)^(1/4)*d^(7/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4
*a*c)^(1/4)*Sqrt[d])]

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Rubi [A]  time = 0.318056, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -10 c d^{7/2} \sqrt [4]{b^2-4 a c} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-10 c d^{7/2} \sqrt [4]{b^2-4 a c} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-\frac{d (b d+2 c d x)^{5/2}}{a+b x+c x^2}+20 c d^3 \sqrt{b d+2 c d x} \]

Antiderivative was successfully verified.

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

[Out]

20*c*d^3*Sqrt[b*d + 2*c*d*x] - (d*(b*d + 2*c*d*x)^(5/2))/(a + b*x + c*x^2) - 10*
c*(b^2 - 4*a*c)^(1/4)*d^(7/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sq
rt[d])] - 10*c*(b^2 - 4*a*c)^(1/4)*d^(7/2)*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 = 77.0156, size = 150, normalized size = 1. \[ - 10 c d^{\frac{7}{2}} \sqrt [4]{- 4 a c + b^{2}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 10 c d^{\frac{7}{2}} \sqrt [4]{- 4 a c + b^{2}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + 20 c d^{3} \sqrt{b d + 2 c d x} - \frac{d \left (b d + 2 c d x\right )^{\frac{5}{2}}}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*c*d**(7/2)*(-4*a*c + b**2)**(1/4)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c
+ b**2)**(1/4))) - 10*c*d**(7/2)*(-4*a*c + b**2)**(1/4)*atanh(sqrt(b*d + 2*c*d*x
)/(sqrt(d)*(-4*a*c + b**2)**(1/4))) + 20*c*d**3*sqrt(b*d + 2*c*d*x) - d*(b*d + 2
*c*d*x)**(5/2)/(a + b*x + c*x**2)

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Mathematica [A]  time = 0.608929, size = 147, normalized size = 0.98 \[ (d (b+2 c x))^{7/2} \left (\frac{\frac{4 a c-b^2}{a+x (b+c x)}+16 c}{(b+2 c x)^3}-\frac{10 c \sqrt [4]{b^2-4 a c} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{7/2}}-\frac{10 c \sqrt [4]{b^2-4 a c} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{7/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(d*(b + 2*c*x))^(7/2)*((16*c + (-b^2 + 4*a*c)/(a + x*(b + c*x)))/(b + 2*c*x)^3 -
(10*c*(b^2 - 4*a*c)^(1/4)*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b + 2*c
*x)^(7/2) - (10*c*(b^2 - 4*a*c)^(1/4)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4
)])/(b + 2*c*x)^(7/2))

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Maple [B]  time = 0.02, size = 693, normalized size = 4.6 \[ 16\,c{d}^{3}\sqrt{2\,cdx+bd}+16\,{\frac{{c}^{2}{d}^{5}a\sqrt{2\,cdx+bd}}{4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2}}}-4\,{\frac{c{d}^{5}{b}^{2}\sqrt{2\,cdx+bd}}{4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2}}}+20\,{\frac{{c}^{2}{d}^{5}\sqrt{2}a}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{3/4}}\arctan \left ( -{\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }-5\,{\frac{c{d}^{5}\sqrt{2}{b}^{2}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{3/4}}\arctan \left ( -{\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }-10\,{\frac{{c}^{2}{d}^{5}\sqrt{2}a}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{3/4}}\ln \left ({\frac{2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}{2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}} \right ) }+{\frac{5\,c{d}^{5}\sqrt{2}{b}^{2}}{2}\ln \left ({1 \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}-20\,{\frac{{c}^{2}{d}^{5}\sqrt{2}a}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{3/4}}\arctan \left ({\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }+5\,{\frac{c{d}^{5}\sqrt{2}{b}^{2}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{3/4}}\arctan \left ({\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16*c*d^3*(2*c*d*x+b*d)^(1/2)+16*c^2*d^5*(2*c*d*x+b*d)^(1/2)/(4*c^2*d^2*x^2+4*b*c
*d^2*x+4*a*c*d^2)*a-4*c*d^5*(2*c*d*x+b*d)^(1/2)/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c
*d^2)*b^2+20*c^2*d^5/(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)*a-5*c*d^5/(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)*b^2-10*c^2*
d^5/(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)))*a+5/2*c*d^
5/(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)))*b^2-20*c^2*d
^5/(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)*a+5*c*d^5/(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)*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)^(7/2)/(c*x^2 + b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232329, size = 429, normalized size = 2.86 \[ \frac{20 \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac{1}{4}}{\left (c x^{2} + b x + a\right )} \arctan \left (\frac{\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac{1}{4}}}{\sqrt{2 \, c d x + b d} c d^{3} + \sqrt{2 \, c^{3} d^{7} x + b c^{2} d^{7} + \sqrt{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}}}}\right ) - 5 \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac{1}{4}}{\left (c x^{2} + b x + a\right )} \log \left (5 \, \sqrt{2 \, c d x + b d} c d^{3} + 5 \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac{1}{4}}\right ) + 5 \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac{1}{4}}{\left (c x^{2} + b x + a\right )} \log \left (5 \, \sqrt{2 \, c d x + b d} c d^{3} - 5 \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac{1}{4}}\right ) +{\left (16 \, c^{2} d^{3} x^{2} + 16 \, b c d^{3} x -{\left (b^{2} - 20 \, a c\right )} d^{3}\right )} \sqrt{2 \, c d x + b d}}{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(20*((b^2*c^4 - 4*a*c^5)*d^14)^(1/4)*(c*x^2 + b*x + a)*arctan(((b^2*c^4 - 4*a*c^
5)*d^14)^(1/4)/(sqrt(2*c*d*x + b*d)*c*d^3 + sqrt(2*c^3*d^7*x + b*c^2*d^7 + sqrt(
(b^2*c^4 - 4*a*c^5)*d^14)))) - 5*((b^2*c^4 - 4*a*c^5)*d^14)^(1/4)*(c*x^2 + b*x +
a)*log(5*sqrt(2*c*d*x + b*d)*c*d^3 + 5*((b^2*c^4 - 4*a*c^5)*d^14)^(1/4)) + 5*((
b^2*c^4 - 4*a*c^5)*d^14)^(1/4)*(c*x^2 + b*x + a)*log(5*sqrt(2*c*d*x + b*d)*c*d^3
- 5*((b^2*c^4 - 4*a*c^5)*d^14)^(1/4)) + (16*c^2*d^3*x^2 + 16*b*c*d^3*x - (b^2 -
20*a*c)*d^3)*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)**(7/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.24115, size = 595, normalized size = 3.97 \[ -5 \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c d^{3} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) - 5 \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c d^{3} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) - \frac{5}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c d^{3}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac{5}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c d^{3}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + 16 \, \sqrt{2 \, c d x + b d} c d^{3} + \frac{4 \,{\left (\sqrt{2 \, c d x + b d} b^{2} c d^{5} - 4 \, \sqrt{2 \, c d x + b d} a c^{2} d^{5}\right )}}{b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c*d^3*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)) -
5*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c*d^3*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)) -
5/2*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c*d^3*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)) + 5/2*s
qrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c*d^3*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)) + 16*sqrt(2*
c*d*x + b*d)*c*d^3 + 4*(sqrt(2*c*d*x + b*d)*b^2*c*d^5 - 4*sqrt(2*c*d*x + b*d)*a*
c^2*d^5)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)