3.1276 \(\int \frac{(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=145 \[ -2 d^{7/2} \left (b^2-4 a c\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{7/2} \left (b^2-4 a c\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+4 d^3 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}+\frac{4}{5} d (b d+2 c d x)^{5/2} \]

[Out]

4*(b^2 - 4*a*c)*d^3*Sqrt[b*d + 2*c*d*x] + (4*d*(b*d + 2*c*d*x)^(5/2))/5 - 2*(b^2
- 4*a*c)^(5/4)*d^(7/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])
] - 2*(b^2 - 4*a*c)^(5/4)*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 = 0.325157, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -2 d^{7/2} \left (b^2-4 a c\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{7/2} \left (b^2-4 a c\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+4 d^3 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}+\frac{4}{5} d (b d+2 c d x)^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

4*(b^2 - 4*a*c)*d^3*Sqrt[b*d + 2*c*d*x] + (4*d*(b*d + 2*c*d*x)^(5/2))/5 - 2*(b^2
- 4*a*c)^(5/4)*d^(7/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])
] - 2*(b^2 - 4*a*c)^(5/4)*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 = 77.1205, size = 146, normalized size = 1.01 \[ - 2 d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 2 d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + 4 d^{3} \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} + \frac{4 d \left (b d + 2 c d x\right )^{\frac{5}{2}}}{5} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.327248, size = 141, normalized size = 0.97 \[ \frac{2 d^3 \sqrt{d (b+2 c x)} \left (4 \sqrt{b+2 c x} \left (2 c \left (c x^2-5 a\right )+3 b^2+2 b c x\right )-5 \left (b^2-4 a c\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-5 \left (b^2-4 a c\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )}{5 \sqrt{b+2 c x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 922, normalized size = 6.4 \[ \text{result too large to display} \]

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),x)

[Out]

4/5*d*(2*c*d*x+b*d)^(5/2)-16*a*c*d^3*(2*c*d*x+b*d)^(1/2)+4*b^2*d^3*(2*c*d*x+b*d)
^(1/2)-16*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^2*c^2+8*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*b^2*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^4+8*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^2*c^2-4*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*b^2*c+1/2*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)))*b^4+16*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^2*c^2-8*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*b^
2*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^4

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23833, size = 810, normalized size = 5.59 \[ \frac{8}{5} \,{\left (2 \, c^{2} d^{3} x^{2} + 2 \, b c d^{3} x +{\left (3 \, b^{2} - 10 \, a c\right )} d^{3}\right )} \sqrt{2 \, c d x + b d} - 4 \, \left ({\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} d^{14}\right )^{\frac{1}{4}} \arctan \left (-\frac{\left ({\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} d^{14}\right )^{\frac{1}{4}}}{\sqrt{2 \, c d x + b d}{\left (b^{2} - 4 \, a c\right )} d^{3} - \sqrt{2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{7} x +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{7} + \sqrt{{\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} d^{14}}}}\right ) + \left ({\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} d^{14}\right )^{\frac{1}{4}} \log \left (-\sqrt{2 \, c d x + b d}{\left (b^{2} - 4 \, a c\right )} d^{3} + \left ({\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} d^{14}\right )^{\frac{1}{4}}\right ) - \left ({\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} d^{14}\right )^{\frac{1}{4}} \log \left (-\sqrt{2 \, c d x + b d}{\left (b^{2} - 4 \, a c\right )} d^{3} - \left ({\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} d^{14}\right )^{\frac{1}{4}}\right ) \]

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),x, algorithm="fricas")

[Out]

8/5*(2*c^2*d^3*x^2 + 2*b*c*d^3*x + (3*b^2 - 10*a*c)*d^3)*sqrt(2*c*d*x + b*d) - 4
*((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 - 10
24*a^5*c^5)*d^14)^(1/4)*arctan(-((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^14)^(1/4)/(sqrt(2*c*d*x + b*d)*(b^2
- 4*a*c)*d^3 - sqrt(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^7*x + (b^5 - 8*a*b^3
*c + 16*a^2*b*c^2)*d^7 + sqrt((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^14)))) + ((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^14)^(1/4)*log(-s
qrt(2*c*d*x + b*d)*(b^2 - 4*a*c)*d^3 + ((b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 6
40*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*d^14)^(1/4)) - ((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^14
)^(1/4)*log(-sqrt(2*c*d*x + b*d)*(b^2 - 4*a*c)*d^3 - ((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^14)^(1/4))

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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)**(7/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.247487, size = 612, normalized size = 4.22 \[ 4 \, \sqrt{2 \, c d x + b d} b^{2} d^{3} - 16 \, \sqrt{2 \, c d x + b d} a c d^{3} + \frac{4}{5} \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} d - \frac{1}{2} \, \sqrt{2}{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac{1}{2} \, \sqrt{2}{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) -{\left (\sqrt{2} b^{2} d^{3} - 4 \, \sqrt{2} a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) -{\left (\sqrt{2} b^{2} d^{3} - 4 \, \sqrt{2} a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) \]

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),x, algorithm="giac")

[Out]

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