3.1368 \(\int \frac{(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=205 \[ \frac{60 d^{11/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{7 \sqrt{a+b x+c x^2}}+\frac{120}{7} c d^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{72}{7} c d^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac{2 d (b d+2 c d x)^{9/2}}{\sqrt{a+b x+c x^2}} \]

[Out]

(-2*d*(b*d + 2*c*d*x)^(9/2))/Sqrt[a + b*x + c*x^2] + (120*c*(b^2 - 4*a*c)*d^5*Sq
rt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/7 + (72*c*d^3*(b*d + 2*c*d*x)^(5/2)*Sqr
t[a + b*x + c*x^2])/7 + (60*(b^2 - 4*a*c)^(9/4)*d^(11/2)*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])/(7*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.509142, antiderivative size = 205, 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 \[ \frac{60 d^{11/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{7 \sqrt{a+b x+c x^2}}+\frac{120}{7} c d^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{72}{7} c d^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac{2 d (b d+2 c d x)^{9/2}}{\sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d*(b*d + 2*c*d*x)^(9/2))/Sqrt[a + b*x + c*x^2] + (120*c*(b^2 - 4*a*c)*d^5*Sq
rt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/7 + (72*c*d^3*(b*d + 2*c*d*x)^(5/2)*Sqr
t[a + b*x + c*x^2])/7 + (60*(b^2 - 4*a*c)^(9/4)*d^(11/2)*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])/(7*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 108.864, size = 202, normalized size = 0.99 \[ \frac{120 c d^{5} \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{7} + \frac{72 c d^{3} \left (b d + 2 c d x\right )^{\frac{5}{2}} \sqrt{a + b x + c x^{2}}}{7} + \frac{60 d^{\frac{11}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{7 \sqrt{a + b x + c x^{2}}} - \frac{2 d \left (b d + 2 c d x\right )^{\frac{9}{2}}}{\sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

120*c*d**5*(-4*a*c + b**2)*sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)/7 + 72*c*d
**3*(b*d + 2*c*d*x)**(5/2)*sqrt(a + b*x + c*x**2)/7 + 60*d**(11/2)*sqrt(c*(a + b
*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(9/4)*elliptic_f(asin(sqrt(b*d + 2
*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(7*sqrt(a + b*x + c*x**2)) - 2*d*
(b*d + 2*c*d*x)**(9/2)/sqrt(a + b*x + c*x**2)

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Mathematica [C]  time = 1.50466, size = 202, normalized size = 0.99 \[ \frac{(d (b+2 c x))^{11/2} \left (\frac{2 (a+x (b+c x)) \left (-\frac{7 \left (b^2-4 a c\right )^2}{a+x (b+c x)}-8 c \left (16 a c-5 b^2\right )+32 b c^2 x+32 c^3 x^2\right )}{(b+2 c x)^5}+\frac{60 i \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}} (b+2 c x)^{9/2}}\right )}{7 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(11/2)*((2*(a + x*(b + c*x))*(-8*c*(-5*b^2 + 16*a*c) + 32*b*c^2
*x + 32*c^3*x^2 - (7*(b^2 - 4*a*c)^2)/(a + x*(b + c*x))))/(b + 2*c*x)^5 + ((60*I
)*(b^2 - 4*a*c)^2*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)^(9/2))))/(7*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.081, size = 569, normalized size = 2.8 \[{\frac{2\,{d}^{5}}{14\,{x}^{3}{c}^{2}+21\,{x}^{2}bc+14\,acx+7\,{b}^{2}x+7\,ab}\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 64\,{c}^{5}{x}^{5}+240\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{a}^{2}{c}^{2}-120\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}a{b}^{2}c+15\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{b}^{4}+160\,b{c}^{4}{x}^{4}-192\,{x}^{3}a{c}^{4}+208\,{x}^{3}{b}^{2}{c}^{3}-288\,{x}^{2}ab{c}^{3}+152\,{x}^{2}{b}^{3}{c}^{2}-480\,{a}^{2}{c}^{3}x+96\,a{b}^{2}{c}^{2}x+26\,{b}^{4}cx-240\,{a}^{2}b{c}^{2}+96\,ac{b}^{3}-7\,{b}^{5} \right ) } \]

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)^(3/2),x)

[Out]

2/7*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^5*(64*c^5*x^5+240*((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*c^2-120*((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^2*c+15*((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^4+160*b*c^4*x^4-192*
x^3*a*c^4+208*x^3*b^2*c^3-288*x^2*a*b*c^3+152*x^2*b^3*c^2-480*a^2*c^3*x+96*a*b^2
*c^2*x+26*b^4*c*x-240*a^2*b*c^2+96*a*c*b^3-7*b^5)/(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. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (32 \, c^{5} d^{5} x^{5} + 80 \, b c^{4} d^{5} x^{4} + 80 \, b^{2} c^{3} d^{5} x^{3} + 40 \, b^{3} c^{2} d^{5} x^{2} + 10 \, b^{4} c d^{5} x + b^{5} d^{5}\right )} \sqrt{2 \, c d x + b d}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}, x\right ) \]

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)^(3/2),x, algorithm="fricas")

[Out]

integral((32*c^5*d^5*x^5 + 80*b*c^4*d^5*x^4 + 80*b^2*c^3*d^5*x^3 + 40*b^3*c^2*d^
5*x^2 + 10*b^4*c*d^5*x + b^5*d^5)*sqrt(2*c*d*x + b*d)/(c*x^2 + b*x + a)^(3/2), 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)**(11/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

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)^(3/2),x, algorithm="giac")

[Out]

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