3.1316 \(\int (b d+2 c d x)^{7/2} \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=227 \[ -\frac{5 d^{7/2} \left (b^2-4 a c\right )^{13/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{231 c^2 \sqrt{a+b x+c x^2}}-\frac{10 d^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{231 c}-\frac{2 d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{77 c}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d} \]

[Out]

(-10*(b^2 - 4*a*c)^2*d^3*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(231*c) - (2
*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(77*c) + ((b*d + 2
*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2])/(11*c*d) - (5*(b^2 - 4*a*c)^(13/4)*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])/(231*c^2*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.54784, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{5 d^{7/2} \left (b^2-4 a c\right )^{13/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{231 c^2 \sqrt{a+b x+c x^2}}-\frac{10 d^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{231 c}-\frac{2 d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{77 c}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d} \]

Antiderivative was successfully verified.

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

[Out]

(-10*(b^2 - 4*a*c)^2*d^3*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(231*c) - (2
*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(77*c) + ((b*d + 2
*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2])/(11*c*d) - (5*(b^2 - 4*a*c)^(13/4)*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])/(231*c^2*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 110.676, size = 216, normalized size = 0.95 \[ - \frac{10 d^{3} \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{231 c} - \frac{2 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}} \sqrt{a + b x + c x^{2}}}{77 c} + \frac{\left (b d + 2 c d x\right )^{\frac{9}{2}} \sqrt{a + b x + c x^{2}}}{11 c d} - \frac{5 d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{13}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{231 c^{2} \sqrt{a + b x + c x^{2}}} \]

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)**(1/2),x)

[Out]

-10*d**3*(-4*a*c + b**2)**2*sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)/(231*c) -
2*d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(5/2)*sqrt(a + b*x + c*x**2)/(77*c) + (b*d
+ 2*c*d*x)**(9/2)*sqrt(a + b*x + c*x**2)/(11*c*d) - 5*d**(7/2)*sqrt(c*(a + b*x
+ c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(13/4)*elliptic_f(asin(sqrt(b*d + 2*c
*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(231*c**2*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 1.52204, size = 223, normalized size = 0.98 \[ \frac{(d (b+2 c x))^{7/2} \left (\frac{c (a+x (b+c x)) \left (16 c^2 \left (-10 a^2+6 a c x^2+21 c^2 x^4\right )+8 b^2 c \left (13 a+60 c x^2\right )+96 b c^2 x \left (a+7 c x^2\right )+5 b^4+144 b^3 c x\right )}{(b+2 c x)^3}-\frac{5 i \left (b^2-4 a c\right )^3 \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}} (b+2 c x)^{5/2}}\right )}{231 c^2 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(7/2)*((c*(a + x*(b + c*x))*(5*b^4 + 144*b^3*c*x + 96*b*c^2*x*(
a + 7*c*x^2) + 8*b^2*c*(13*a + 60*c*x^2) + 16*c^2*(-10*a^2 + 6*a*c*x^2 + 21*c^2*
x^4)))/(b + 2*c*x)^3 - ((5*I)*(b^2 - 4*a*c)^3*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])/(Sqr
t[-Sqrt[b^2 - 4*a*c]]*(b + 2*c*x)^(5/2))))/(231*c^2*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.183, size = 798, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(1/2),x)

[Out]

1/462*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^3*(1344*c^7*x^7+4704*b*c^6*x^6+1
728*x^5*a*c^6+6624*x^5*b^2*c^5+320*(-4*a*c+b^2)^(1/2)*((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))*a^3*c^3-240*(-4*a*c+b^2)^(1/
2)*((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)*Ellipt
icF(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))
*a^2*b^2*c^2+60*(-4*a*c+b^2)^(1/2)*((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))*a*b^4*c-5*(-4*a*c+b^2)^(1/2)*((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))*b^6+4320*x^4*a*b*c^5
+4800*x^4*b^3*c^4-256*x^3*a^2*c^5+4448*x^3*a*b^2*c^4+1844*x^3*b^4*c^3-384*x^2*a^
2*b*c^4+2352*a*b^3*c^3*x^2+318*x^2*b^5*c^2-640*a^3*c^4*x+288*a^2*b^2*c^3*x+516*c
^2*a*b^4*x+10*b^6*c*x-320*a^3*b*c^3+208*a^2*b^3*c^2+10*a*b^5*c)/c^2/(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. \[ \int{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} \sqrt{c x^{2} + b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((8*c^3*d^3*x^3 + 12*b*c^2*d^3*x^2 + 6*b^2*c*d^3*x + b^3*d^3)*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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.902446, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done