3.1338 \(\int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=321 \[ -\frac{5 d^{7/2} \left (b^2-4 a c\right )^{21/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{17556 c^4 \sqrt{a+b x+c x^2}}-\frac{5 d^3 \left (b^2-4 a c\right )^4 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{8778 c^3}-\frac{d \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{2926 c^3}+\frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{9/2}}{836 c^3 d}-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{114 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{9/2}}{19 c d} \]

[Out]

(-5*(b^2 - 4*a*c)^4*d^3*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(8778*c^3) -
((b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(2926*c^3) + ((b
^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2])/(836*c^3*d) - ((b^2 -
4*a*c)*(b*d + 2*c*d*x)^(9/2)*(a + b*x + c*x^2)^(3/2))/(114*c^2*d) + ((b*d + 2*c
*d*x)^(9/2)*(a + b*x + c*x^2)^(5/2))/(19*c*d) - (5*(b^2 - 4*a*c)^(21/4)*d^(7/2)*
Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x
]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(17556*c^4*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.83608, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{5 d^{7/2} \left (b^2-4 a c\right )^{21/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{17556 c^4 \sqrt{a+b x+c x^2}}-\frac{5 d^3 \left (b^2-4 a c\right )^4 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{8778 c^3}-\frac{d \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{2926 c^3}+\frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{9/2}}{836 c^3 d}-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{114 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{9/2}}{19 c d} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(-5*(b^2 - 4*a*c)^4*d^3*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(8778*c^3) -
((b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(2926*c^3) + ((b
^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2])/(836*c^3*d) - ((b^2 -
4*a*c)*(b*d + 2*c*d*x)^(9/2)*(a + b*x + c*x^2)^(3/2))/(114*c^2*d) + ((b*d + 2*c
*d*x)^(9/2)*(a + b*x + c*x^2)^(5/2))/(19*c*d) - (5*(b^2 - 4*a*c)^(21/4)*d^(7/2)*
Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x
]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(17556*c^4*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 161.526, size = 306, normalized size = 0.95 \[ \frac{\left (b d + 2 c d x\right )^{\frac{9}{2}} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{19 c d} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{9}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{114 c^{2} d} - \frac{5 d^{3} \left (- 4 a c + b^{2}\right )^{4} \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{8778 c^{3}} - \frac{d \left (- 4 a c + b^{2}\right )^{3} \left (b d + 2 c d x\right )^{\frac{5}{2}} \sqrt{a + b x + c x^{2}}}{2926 c^{3}} + \frac{\left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{9}{2}} \sqrt{a + b x + c x^{2}}}{836 c^{3} d} - \frac{5 d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{21}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{17556 c^{4} \sqrt{a + b x + c x^{2}}} \]

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)**(5/2),x)

[Out]

(b*d + 2*c*d*x)**(9/2)*(a + b*x + c*x**2)**(5/2)/(19*c*d) - (-4*a*c + b**2)*(b*d
+ 2*c*d*x)**(9/2)*(a + b*x + c*x**2)**(3/2)/(114*c**2*d) - 5*d**3*(-4*a*c + b**
2)**4*sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)/(8778*c**3) - d*(-4*a*c + b**2)
**3*(b*d + 2*c*d*x)**(5/2)*sqrt(a + b*x + c*x**2)/(2926*c**3) + (-4*a*c + b**2)*
*2*(b*d + 2*c*d*x)**(9/2)*sqrt(a + b*x + c*x**2)/(836*c**3*d) - 5*d**(7/2)*sqrt(
c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(21/4)*elliptic_f(asin(sqr
t(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(17556*c**4*sqrt(a + b*x
+ c*x**2))

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Mathematica [C]  time = 2.73449, size = 386, normalized size = 1.2 \[ \frac{(d (b+2 c x))^{7/2} \left (\frac{c (a+x (b+c x)) \left (4 b^4 c^2 \left (157 a^2+3684 a c x^2+7399 c^2 x^4\right )+32 b^3 c^3 x \left (433 a^2+2142 a c x^2+2310 c^2 x^4\right )+32 b^2 c^3 \left (92 a^3+1371 a^2 c x^2+4151 a c^2 x^4+2926 c^3 x^6\right )+128 b c^4 x \left (12 a^3+469 a^2 c x^2+924 a c^2 x^4+462 c^3 x^6\right )+64 c^4 \left (-40 a^4+24 a^3 c x^2+469 a^2 c^2 x^4+616 a c^3 x^6+231 c^4 x^8\right )+18 b^6 c \left (c x^2-5 a\right )+8 b^5 c^2 x \left (22 a+623 c x^2\right )+5 b^8-10 b^7 c x\right )}{(b+2 c x)^3}-\frac{5 i \left (b^2-4 a c\right )^5 \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}} (b+2 c x)^{5/2}}\right )}{17556 c^4 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2),x]

[Out]

((d*(b + 2*c*x))^(7/2)*((c*(a + x*(b + c*x))*(5*b^8 - 10*b^7*c*x + 18*b^6*c*(-5*
a + c*x^2) + 8*b^5*c^2*x*(22*a + 623*c*x^2) + 32*b^3*c^3*x*(433*a^2 + 2142*a*c*x
^2 + 2310*c^2*x^4) + 4*b^4*c^2*(157*a^2 + 3684*a*c*x^2 + 7399*c^2*x^4) + 128*b*c
^4*x*(12*a^3 + 469*a^2*c*x^2 + 924*a*c^2*x^4 + 462*c^3*x^6) + 32*b^2*c^3*(92*a^3
+ 1371*a^2*c*x^2 + 4151*a*c^2*x^4 + 2926*c^3*x^6) + 64*c^4*(-40*a^4 + 24*a^3*c*
x^2 + 469*a^2*c^2*x^4 + 616*a*c^3*x^6 + 231*c^4*x^8)))/(b + 2*c*x)^3 - ((5*I)*(b
^2 - 4*a*c)^5*Sqrt[(c*(a + x*(b + c*x)))/(b + 2*c*x)^2]*EllipticF[I*ArcSinh[Sqrt
[-Sqrt[b^2 - 4*a*c]]/Sqrt[b + 2*c*x]], -1])/(Sqrt[-Sqrt[b^2 - 4*a*c]]*(b + 2*c*x
)^(5/2))))/(17556*c^4*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.087, size = 1344, normalized size = 4.2 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

1/35112*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^3*(325248*x^10*b*c^10+216832*x
^9*a*c^10+758912*x^9*b^2*c^9+975744*x^8*b^3*c^8+277760*x^7*a^2*c^9+749168*x^7*b^
4*c^7+345352*x^6*b^5*c^6+126208*x^5*a^3*c^8+89168*x^5*b^6*c^5+10036*x^4*b^7*c^4-
4096*x^3*a^4*c^7-4*x^3*b^8*c^3+10*x^2*b^9*c^2-10240*x*a^5*c^6+10*x*b^10*c-5*((b+
2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/
2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2
*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*
c+b^2)^(1/2)*b^10-5120*a^5*b*c^5+5888*a^4*b^3*c^4+1256*a^3*b^5*c^3-180*a^2*b^7*c
^2+10*a*b^9*c-6400*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*
c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/
2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*
2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*a^4*b^2*c^4-180*x*a*b^8*c^2+9728*x*a^4*b^2*c
^5+36112*x*a^3*b^4*c^4+1790656*x^6*a*b^3*c^7+1363584*x^5*a^2*b^2*c^7+1002096*x^5
*a*b^4*c^6+315520*x^4*a^3*b*c^7+319616*x^3*a^3*b^2*c^6+1248*x*a^2*b^6*c^3-6144*x
^2*a^4*b*c^6+163904*x^2*a^3*b^3*c^5+369424*x^3*a^2*b^4*c^5+61656*x^2*a^2*b^5*c^4
+40208*x^3*a*b^6*c^4+1812608*x^7*a*b^2*c^8+972160*x^6*a^2*b*c^8-192*x^2*a*b^7*c^
3+975744*x^8*a*b*c^9+978560*x^4*a^2*b^3*c^6+305336*x^4*a*b^5*c^5+59136*x^11*c^11
+3200*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*
c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*Ell
ipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/
2))*(-4*a*c+b^2)^(1/2)*a^3*b^4*c^3-800*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2
)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/
2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*
c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*a^2*b^6*c^2+100*((b+2*c*
x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^
(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b
+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^
2)^(1/2)*a*b^8*c+5120*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-
(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^
(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/
2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*a^5*c^5)/c^4/(2*c^2*x^3+3*b*c*x^2+2*a*c*x
+b^2*x+a*b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}\,{d x} \]

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)^(5/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (8 \, c^{5} d^{3} x^{7} + 28 \, b c^{4} d^{3} x^{6} + 2 \,{\left (19 \, b^{2} c^{3} + 8 \, a c^{4}\right )} d^{3} x^{5} + a^{2} b^{3} d^{3} + 5 \,{\left (5 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d^{3} x^{4} + 4 \,{\left (2 \, b^{4} c + 9 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d^{3} x^{3} +{\left (b^{5} + 14 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} d^{3} x^{2} + 2 \,{\left (a b^{4} + 3 \, a^{2} b^{2} c\right )} d^{3} x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}, x\right ) \]

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)^(5/2),x, algorithm="fricas")

[Out]

integral((8*c^5*d^3*x^7 + 28*b*c^4*d^3*x^6 + 2*(19*b^2*c^3 + 8*a*c^4)*d^3*x^5 +
a^2*b^3*d^3 + 5*(5*b^3*c^2 + 8*a*b*c^3)*d^3*x^4 + 4*(2*b^4*c + 9*a*b^2*c^2 + 2*a
^2*c^3)*d^3*x^3 + (b^5 + 14*a*b^3*c + 12*a^2*b*c^2)*d^3*x^2 + 2*(a*b^4 + 3*a^2*b
^2*c)*d^3*x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a), x)

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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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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)^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError