3.1270 \(\int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{7/2}} \, dx\)

Optimal. Leaf size=121 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{64 c^4 d^5}-\frac{3 \left (b^2-4 a c\right )^2}{64 c^4 d^3 \sqrt{b d+2 c d x}}+\frac{\left (b^2-4 a c\right )^3}{320 c^4 d (b d+2 c d x)^{5/2}}+\frac{(b d+2 c d x)^{7/2}}{448 c^4 d^7} \]

[Out]

(b^2 - 4*a*c)^3/(320*c^4*d*(b*d + 2*c*d*x)^(5/2)) - (3*(b^2 - 4*a*c)^2)/(64*c^4*
d^3*Sqrt[b*d + 2*c*d*x]) - ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2))/(64*c^4*d^5) +
(b*d + 2*c*d*x)^(7/2)/(448*c^4*d^7)

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Rubi [A]  time = 0.141587, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{64 c^4 d^5}-\frac{3 \left (b^2-4 a c\right )^2}{64 c^4 d^3 \sqrt{b d+2 c d x}}+\frac{\left (b^2-4 a c\right )^3}{320 c^4 d (b d+2 c d x)^{5/2}}+\frac{(b d+2 c d x)^{7/2}}{448 c^4 d^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(7/2),x]

[Out]

(b^2 - 4*a*c)^3/(320*c^4*d*(b*d + 2*c*d*x)^(5/2)) - (3*(b^2 - 4*a*c)^2)/(64*c^4*
d^3*Sqrt[b*d + 2*c*d*x]) - ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2))/(64*c^4*d^5) +
(b*d + 2*c*d*x)^(7/2)/(448*c^4*d^7)

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Rubi in Sympy [A]  time = 35.8651, size = 116, normalized size = 0.96 \[ \frac{\left (- 4 a c + b^{2}\right )^{3}}{320 c^{4} d \left (b d + 2 c d x\right )^{\frac{5}{2}}} - \frac{3 \left (- 4 a c + b^{2}\right )^{2}}{64 c^{4} d^{3} \sqrt{b d + 2 c d x}} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}}}{64 c^{4} d^{5}} + \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}}}{448 c^{4} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**(7/2),x)

[Out]

(-4*a*c + b**2)**3/(320*c**4*d*(b*d + 2*c*d*x)**(5/2)) - 3*(-4*a*c + b**2)**2/(6
4*c**4*d**3*sqrt(b*d + 2*c*d*x)) - (-4*a*c + b**2)*(b*d + 2*c*d*x)**(3/2)/(64*c*
*4*d**5) + (b*d + 2*c*d*x)**(7/2)/(448*c**4*d**7)

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Mathematica [A]  time = 0.378982, size = 107, normalized size = 0.88 \[ \frac{(b+2 c x)^4 \left (-40 c x \left (b^2-7 a c\right )-\frac{105 \left (b^2-4 a c\right )^2}{b+2 c x}+\frac{7 \left (b^2-4 a c\right )^3}{(b+2 c x)^3}+140 a b c-30 b^3+60 b c^2 x^2+40 c^3 x^3\right )}{2240 c^4 (d (b+2 c x))^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(7/2),x]

[Out]

((b + 2*c*x)^4*(-30*b^3 + 140*a*b*c - 40*c*(b^2 - 7*a*c)*x + 60*b*c^2*x^2 + 40*c
^3*x^3 + (7*(b^2 - 4*a*c)^3)/(b + 2*c*x)^3 - (105*(b^2 - 4*a*c)^2)/(b + 2*c*x)))
/(2240*c^4*(d*(b + 2*c*x))^(7/2))

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Maple [A]  time = 0.011, size = 174, normalized size = 1.4 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( -5\,{c}^{6}{x}^{6}-15\,b{c}^{5}{x}^{5}-35\,a{c}^{5}{x}^{4}-10\,{b}^{2}{c}^{4}{x}^{4}-70\,ab{c}^{4}{x}^{3}+5\,{b}^{3}{c}^{3}{x}^{3}+105\,{a}^{2}{c}^{4}{x}^{2}-105\,a{b}^{2}{c}^{3}{x}^{2}+15\,{b}^{4}{c}^{2}{x}^{2}+105\,{a}^{2}b{c}^{3}x-70\,a{b}^{3}{c}^{2}x+10\,{b}^{5}cx+7\,{a}^{3}{c}^{3}+21\,{a}^{2}{b}^{2}{c}^{2}-14\,a{b}^{4}c+2\,{b}^{6} \right ) }{35\,{c}^{4}} \left ( 2\,cdx+bd \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(7/2),x)

[Out]

-1/35*(2*c*x+b)*(-5*c^6*x^6-15*b*c^5*x^5-35*a*c^5*x^4-10*b^2*c^4*x^4-70*a*b*c^4*
x^3+5*b^3*c^3*x^3+105*a^2*c^4*x^2-105*a*b^2*c^3*x^2+15*b^4*c^2*x^2+105*a^2*b*c^3
*x-70*a*b^3*c^2*x+10*b^5*c*x+7*a^3*c^3+21*a^2*b^2*c^2-14*a*b^4*c+2*b^6)/c^4/(2*c
*d*x+b*d)^(7/2)

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Maxima [A]  time = 0.688839, size = 192, normalized size = 1.59 \[ -\frac{\frac{7 \,{\left (15 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{2} -{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{2}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{3} d^{2}} + \frac{5 \,{\left (7 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} -{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}\right )}}{c^{3} d^{6}}}{2240 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(2*c*d*x + b*d)^(7/2),x, algorithm="maxima")

[Out]

-1/2240*(7*(15*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*(2*c*d*x + b*d)^2 - (b^6 - 12*a*b^
4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^2)/((2*c*d*x + b*d)^(5/2)*c^3*d^2) + 5*(7*(
2*c*d*x + b*d)^(3/2)*(b^2 - 4*a*c)*d^2 - (2*c*d*x + b*d)^(7/2))/(c^3*d^6))/(c*d)

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Fricas [A]  time = 0.208593, size = 262, normalized size = 2.17 \[ \frac{5 \, c^{6} x^{6} + 15 \, b c^{5} x^{5} - 2 \, b^{6} + 14 \, a b^{4} c - 21 \, a^{2} b^{2} c^{2} - 7 \, a^{3} c^{3} + 5 \,{\left (2 \, b^{2} c^{4} + 7 \, a c^{5}\right )} x^{4} - 5 \,{\left (b^{3} c^{3} - 14 \, a b c^{4}\right )} x^{3} - 15 \,{\left (b^{4} c^{2} - 7 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} - 5 \,{\left (2 \, b^{5} c - 14 \, a b^{3} c^{2} + 21 \, a^{2} b c^{3}\right )} x}{35 \,{\left (4 \, c^{6} d^{3} x^{2} + 4 \, b c^{5} d^{3} x + b^{2} c^{4} d^{3}\right )} \sqrt{2 \, c d x + b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(2*c*d*x + b*d)^(7/2),x, algorithm="fricas")

[Out]

1/35*(5*c^6*x^6 + 15*b*c^5*x^5 - 2*b^6 + 14*a*b^4*c - 21*a^2*b^2*c^2 - 7*a^3*c^3
+ 5*(2*b^2*c^4 + 7*a*c^5)*x^4 - 5*(b^3*c^3 - 14*a*b*c^4)*x^3 - 15*(b^4*c^2 - 7*
a*b^2*c^3 + 7*a^2*c^4)*x^2 - 5*(2*b^5*c - 14*a*b^3*c^2 + 21*a^2*b*c^3)*x)/((4*c^
6*d^3*x^2 + 4*b*c^5*d^3*x + b^2*c^4*d^3)*sqrt(2*c*d*x + b*d))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x + c x^{2}\right )^{3}}{\left (d \left (b + 2 c x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**(7/2),x)

[Out]

Integral((a + b*x + c*x**2)**3/(d*(b + 2*c*x))**(7/2), x)

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GIAC/XCAS [A]  time = 0.239695, size = 251, normalized size = 2.07 \[ \frac{b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2} - 15 \,{\left (2 \, c d x + b d\right )}^{2} b^{4} + 120 \,{\left (2 \, c d x + b d\right )}^{2} a b^{2} c - 240 \,{\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2}}{320 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{4} d^{3}} - \frac{7 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c^{24} d^{44} - 28 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a c^{25} d^{44} -{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{24} d^{42}}{448 \, c^{28} d^{49}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(2*c*d*x + b*d)^(7/2),x, algorithm="giac")

[Out]

1/320*(b^6*d^2 - 12*a*b^4*c*d^2 + 48*a^2*b^2*c^2*d^2 - 64*a^3*c^3*d^2 - 15*(2*c*
d*x + b*d)^2*b^4 + 120*(2*c*d*x + b*d)^2*a*b^2*c - 240*(2*c*d*x + b*d)^2*a^2*c^2
)/((2*c*d*x + b*d)^(5/2)*c^4*d^3) - 1/448*(7*(2*c*d*x + b*d)^(3/2)*b^2*c^24*d^44
- 28*(2*c*d*x + b*d)^(3/2)*a*c^25*d^44 - (2*c*d*x + b*d)^(7/2)*c^24*d^42)/(c^28
*d^49)