3.1299 \(\int \frac{(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=191 \[ -45 c^2 d^{11/2} \sqrt [4]{b^2-4 a c} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-45 c^2 d^{11/2} \sqrt [4]{b^2-4 a c} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-\frac{9 c d^3 (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}+90 c^2 d^5 \sqrt{b d+2 c d x} \]

[Out]

90*c^2*d^5*Sqrt[b*d + 2*c*d*x] - (d*(b*d + 2*c*d*x)^(9/2))/(2*(a + b*x + c*x^2)^
2) - (9*c*d^3*(b*d + 2*c*d*x)^(5/2))/(2*(a + b*x + c*x^2)) - 45*c^2*(b^2 - 4*a*c
)^(1/4)*d^(11/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 45*
c^2*(b^2 - 4*a*c)^(1/4)*d^(11/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4
)*Sqrt[d])]

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Rubi [A]  time = 0.393617, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -45 c^2 d^{11/2} \sqrt [4]{b^2-4 a c} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-45 c^2 d^{11/2} \sqrt [4]{b^2-4 a c} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-\frac{9 c d^3 (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}+90 c^2 d^5 \sqrt{b d+2 c d x} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^3,x]

[Out]

90*c^2*d^5*Sqrt[b*d + 2*c*d*x] - (d*(b*d + 2*c*d*x)^(9/2))/(2*(a + b*x + c*x^2)^
2) - (9*c*d^3*(b*d + 2*c*d*x)^(5/2))/(2*(a + b*x + c*x^2)) - 45*c^2*(b^2 - 4*a*c
)^(1/4)*d^(11/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 45*
c^2*(b^2 - 4*a*c)^(1/4)*d^(11/2)*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 = 95.5505, size = 194, normalized size = 1.02 \[ - 45 c^{2} d^{\frac{11}{2}} \sqrt [4]{- 4 a c + b^{2}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 45 c^{2} d^{\frac{11}{2}} \sqrt [4]{- 4 a c + b^{2}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + 90 c^{2} d^{5} \sqrt{b d + 2 c d x} - \frac{9 c d^{3} \left (b d + 2 c d x\right )^{\frac{5}{2}}}{2 \left (a + b x + c x^{2}\right )} - \frac{d \left (b d + 2 c d x\right )^{\frac{9}{2}}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(11/2)/(c*x**2+b*x+a)**3,x)

[Out]

-45*c**2*d**(11/2)*(-4*a*c + b**2)**(1/4)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*
a*c + b**2)**(1/4))) - 45*c**2*d**(11/2)*(-4*a*c + b**2)**(1/4)*atanh(sqrt(b*d +
2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))) + 90*c**2*d**5*sqrt(b*d + 2*c*d*x) -
9*c*d**3*(b*d + 2*c*d*x)**(5/2)/(2*(a + b*x + c*x**2)) - d*(b*d + 2*c*d*x)**(9/
2)/(2*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 1.18648, size = 182, normalized size = 0.95 \[ (d (b+2 c x))^{11/2} \left (\frac{\frac{17 c \left (4 a c-b^2\right )}{2 (a+x (b+c x))}-\frac{\left (b^2-4 a c\right )^2}{2 (a+x (b+c x))^2}+64 c^2}{(b+2 c x)^5}-\frac{45 c^2 \sqrt [4]{b^2-4 a c} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{11/2}}-\frac{45 c^2 \sqrt [4]{b^2-4 a c} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{11/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^3,x]

[Out]

(d*(b + 2*c*x))^(11/2)*((64*c^2 - (b^2 - 4*a*c)^2/(2*(a + x*(b + c*x))^2) + (17*
c*(-b^2 + 4*a*c))/(2*(a + x*(b + c*x))))/(b + 2*c*x)^5 - (45*c^2*(b^2 - 4*a*c)^(
1/4)*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(11/2) - (45*c^2*(
b^2 - 4*a*c)^(1/4)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(11
/2))

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Maple [B]  time = 0.025, size = 857, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^3,x)

[Out]

64*c^2*d^5*(2*c*d*x+b*d)^(1/2)+136*c^3*d^7/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)
^2*(2*c*d*x+b*d)^(5/2)*a-34*c^2*d^7/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)^2*(2*c
*d*x+b*d)^(5/2)*b^2+416*c^4*d^9/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)^2*(2*c*d*x
+b*d)^(1/2)*a^2-208*c^3*d^9/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)^2*(2*c*d*x+b*d
)^(1/2)*a*b^2+26*c^2*d^9/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)^2*(2*c*d*x+b*d)^(
1/2)*b^4-90*c^3*d^7/(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+45/2*c^2*d^7/(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+90*c^
3*d^7/(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-45/2*c^2*d^7/(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^2-45*c^3*d^7/(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+45/4*c^2*d^7/(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

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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)^(11/2)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238495, size = 652, normalized size = 3.41 \[ \frac{180 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \arctan \left (\frac{\left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac{1}{4}}}{\sqrt{2 \, c d x + b d} c^{2} d^{5} + \sqrt{2 \, c^{5} d^{11} x + b c^{4} d^{11} + \sqrt{{\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}}}}\right ) - 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (45 \, \sqrt{2 \, c d x + b d} c^{2} d^{5} + 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac{1}{4}}\right ) + 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (45 \, \sqrt{2 \, c d x + b d} c^{2} d^{5} - 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac{1}{4}}\right ) +{\left (128 \, c^{4} d^{5} x^{4} + 256 \, b c^{3} d^{5} x^{3} + 3 \,{\left (37 \, b^{2} c^{2} + 108 \, a c^{3}\right )} d^{5} x^{2} -{\left (17 \, b^{3} c - 324 \, a b c^{2}\right )} d^{5} x -{\left (b^{4} + 9 \, a b^{2} c - 180 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt{2 \, c d x + b d}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(11/2)/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

1/2*(180*((b^2*c^8 - 4*a*c^9)*d^22)^(1/4)*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2
+ 2*a*c)*x^2 + a^2)*arctan(((b^2*c^8 - 4*a*c^9)*d^22)^(1/4)/(sqrt(2*c*d*x + b*d)
*c^2*d^5 + sqrt(2*c^5*d^11*x + b*c^4*d^11 + sqrt((b^2*c^8 - 4*a*c^9)*d^22)))) -
45*((b^2*c^8 - 4*a*c^9)*d^22)^(1/4)*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*
c)*x^2 + a^2)*log(45*sqrt(2*c*d*x + b*d)*c^2*d^5 + 45*((b^2*c^8 - 4*a*c^9)*d^22)
^(1/4)) + 45*((b^2*c^8 - 4*a*c^9)*d^22)^(1/4)*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (
b^2 + 2*a*c)*x^2 + a^2)*log(45*sqrt(2*c*d*x + b*d)*c^2*d^5 - 45*((b^2*c^8 - 4*a*
c^9)*d^22)^(1/4)) + (128*c^4*d^5*x^4 + 256*b*c^3*d^5*x^3 + 3*(37*b^2*c^2 + 108*a
*c^3)*d^5*x^2 - (17*b^3*c - 324*a*b*c^2)*d^5*x - (b^4 + 9*a*b^2*c - 180*a^2*c^2)
*d^5)*sqrt(2*c*d*x + b*d))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 +
a^2)

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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)**(11/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.25927, size = 703, normalized size = 3.68 \[ -\frac{45}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{5} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) - \frac{45}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{5} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) - \frac{45}{4} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{5}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac{45}{4} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{5}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + 64 \, \sqrt{2 \, c d x + b d} c^{2} d^{5} + \frac{2 \,{\left (13 \, \sqrt{2 \, c d x + b d} b^{4} c^{2} d^{9} - 104 \, \sqrt{2 \, c d x + b d} a b^{2} c^{3} d^{9} + 208 \, \sqrt{2 \, c d x + b d} a^{2} c^{4} d^{9} - 17 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{2} c^{2} d^{7} + 68 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} a c^{3} d^{7}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(11/2)/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

-45/2*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*d^5*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
)) - 45/2*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*d^5*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)) - 45/4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*d^5*ln(2*c*d*x + b*d + s
qrt(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)) + 45/4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*d^5*ln(2*c*d*x + b*d - sqr
t(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)) + 64*sqrt(2*c*d*x + b*d)*c^2*d^5 + 2*(13*sqrt(2*c*d*x + b*d)*b^4*c^2*d^9 - 1
04*sqrt(2*c*d*x + b*d)*a*b^2*c^3*d^9 + 208*sqrt(2*c*d*x + b*d)*a^2*c^4*d^9 - 17*
(2*c*d*x + b*d)^(5/2)*b^2*c^2*d^7 + 68*(2*c*d*x + b*d)^(5/2)*a*c^3*d^7)/(b^2*d^2
- 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2