3.1378 \(\int \frac{(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=247 \[ \frac{1040}{7} c^2 d^7 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{520 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{7 \sqrt{a+b x+c x^2}}+\frac{624}{7} c^2 d^5 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac{52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*d*(b*d + 2*c*d*x)^(13/2))/(3*(a + b*x + c*x^2)^(3/2)) - (52*c*d^3*(b*d + 2*c
*d*x)^(9/2))/(3*Sqrt[a + b*x + c*x^2]) + (1040*c^2*(b^2 - 4*a*c)*d^7*Sqrt[b*d +
2*c*d*x]*Sqrt[a + b*x + c*x^2])/7 + (624*c^2*d^5*(b*d + 2*c*d*x)^(5/2)*Sqrt[a +
b*x + c*x^2])/7 + (520*c*(b^2 - 4*a*c)^(9/4)*d^(15/2)*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)*Sqr
t[d])], -1])/(7*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.607228, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{1040}{7} c^2 d^7 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{520 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{7 \sqrt{a+b x+c x^2}}+\frac{624}{7} c^2 d^5 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac{52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*d*(b*d + 2*c*d*x)^(13/2))/(3*(a + b*x + c*x^2)^(3/2)) - (52*c*d^3*(b*d + 2*c
*d*x)^(9/2))/(3*Sqrt[a + b*x + c*x^2]) + (1040*c^2*(b^2 - 4*a*c)*d^7*Sqrt[b*d +
2*c*d*x]*Sqrt[a + b*x + c*x^2])/7 + (624*c^2*d^5*(b*d + 2*c*d*x)^(5/2)*Sqrt[a +
b*x + c*x^2])/7 + (520*c*(b^2 - 4*a*c)^(9/4)*d^(15/2)*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)*Sqr
t[d])], -1])/(7*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 125.344, size = 245, normalized size = 0.99 \[ \frac{1040 c^{2} d^{7} \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{7} + \frac{624 c^{2} d^{5} \left (b d + 2 c d x\right )^{\frac{5}{2}} \sqrt{a + b x + c x^{2}}}{7} + \frac{520 c d^{\frac{15}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{7 \sqrt{a + b x + c x^{2}}} - \frac{52 c d^{3} \left (b d + 2 c d x\right )^{\frac{9}{2}}}{3 \sqrt{a + b x + c x^{2}}} - \frac{2 d \left (b d + 2 c d x\right )^{\frac{13}{2}}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(15/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

1040*c**2*d**7*(-4*a*c + b**2)*sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)/7 + 62
4*c**2*d**5*(b*d + 2*c*d*x)**(5/2)*sqrt(a + b*x + c*x**2)/7 + 520*c*d**(15/2)*sq
rt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(9/4)*elliptic_f(asin(s
qrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(7*sqrt(a + b*x + c*x*
*2)) - 52*c*d**3*(b*d + 2*c*d*x)**(9/2)/(3*sqrt(a + b*x + c*x**2)) - 2*d*(b*d +
2*c*d*x)**(13/2)/(3*(a + b*x + c*x**2)**(3/2))

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Mathematica [C]  time = 2.67516, size = 231, normalized size = 0.94 \[ \frac{(d (b+2 c x))^{15/2} \left (\frac{2 (a+x (b+c x)) \left (-64 c^2 \left (38 a c-11 b^2\right )-\frac{266 c \left (b^2-4 a c\right )^2}{a+x (b+c x)}-\frac{7 \left (b^2-4 a c\right )^3}{(a+x (b+c x))^2}+384 b c^3 x+384 c^4 x^2\right )}{3 (b+2 c x)^7}+\frac{520 i c \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}} (b+2 c x)^{13/2}}\right )}{7 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^(5/2),x]

[Out]

((d*(b + 2*c*x))^(15/2)*((2*(a + x*(b + c*x))*(-64*c^2*(-11*b^2 + 38*a*c) + 384*
b*c^3*x + 384*c^4*x^2 - (7*(b^2 - 4*a*c)^3)/(a + x*(b + c*x))^2 - (266*c*(b^2 -
4*a*c)^2)/(a + x*(b + c*x))))/(3*(b + 2*c*x)^7) + ((520*I)*c*(b^2 - 4*a*c)^2*Sqr
t[(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)^(13/2))))/(7*Sq
rt[a + x*(b + c*x)])

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Maple [B]  time = 0.087, size = 1473, normalized size = 6. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^(5/2),x)

[Out]

2/21*(6240*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))*x^2*a^2*c^4*(-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)-6240*a^3*b*c^3+2496*a^2*b^3*c^2-182*a*b^
5*c-3120*(-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^2*b^2*c^2-3328*x^5*a*c^6+4864*x^5*b^2*c^5+5440*x^4*b
^3*c^4-17472*x^3*a^2*c^5+2668*x^3*b^4*c^3-94*x^2*b^5*c^2-12480*a^3*c^4*x-280*b^6
*c*x+390*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))*x^2*b^4*c^2*(-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)+390*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))*x*b^5*c*(-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)-8320*x^4*a*b*
c^5+416*x^3*a*b^2*c^4-26208*x^2*a^2*b*c^4+8944*a*b^3*c^3*x^2-3744*a^2*b^2*c^3*x+
3172*c^2*a*b^4*x+2688*b*c^6*x^6-7*b^7+390*(-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+6240*(-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^3-3120*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))*x^2*a*b^2*c^3*(-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)+6240*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))*x*a^2*b*c^3*(-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)-3120*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))*x*a*b^3*c^2*(-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)+768*c^7*x^7)*d^7*(d*(2*c*x+b))^(1/2)/(2
*c*x+b)/(c*x^2+b*x+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{15}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(15/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(15/2)/(c*x^2 + b*x + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (128 \, c^{7} d^{7} x^{7} + 448 \, b c^{6} d^{7} x^{6} + 672 \, b^{2} c^{5} d^{7} x^{5} + 560 \, b^{3} c^{4} d^{7} x^{4} + 280 \, b^{4} c^{3} d^{7} x^{3} + 84 \, b^{5} c^{2} d^{7} x^{2} + 14 \, b^{6} c d^{7} x + b^{7} d^{7}\right )} \sqrt{2 \, c d x + b d}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(15/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

integral((128*c^7*d^7*x^7 + 448*b*c^6*d^7*x^6 + 672*b^2*c^5*d^7*x^5 + 560*b^3*c^
4*d^7*x^4 + 280*b^4*c^3*d^7*x^3 + 84*b^5*c^2*d^7*x^2 + 14*b^6*c*d^7*x + b^7*d^7)
*sqrt(2*c*d*x + b*d)/((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*
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)**(15/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.657267, 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)^(15/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

Done