3.9 \(\int \frac{A+B x}{\left (a+b x+c x^2\right )^{5/2} \left (d-f x^2\right )} \, dx\)

Optimal. Leaf size=797 \[ -\frac{\left (B \sqrt{d}-A \sqrt{f}\right ) \tanh ^{-1}\left (\frac{-2 \sqrt{f} a+\left (2 c \sqrt{d}-b \sqrt{f}\right ) x+b \sqrt{d}}{2 \sqrt{-\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (-\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}+\frac{\left (\sqrt{f} A+B \sqrt{d}\right ) \tanh ^{-1}\left (\frac{2 \sqrt{f} a+\left (\sqrt{f} b+2 c \sqrt{d}\right ) x+b \sqrt{d}}{2 \sqrt{\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}-\frac{2 \left (3 B d f^2 b^6-A f^2 (7 c d+6 a f) b^5-B f \left (7 c^2 d^2+14 a c f d-3 a^2 f^2\right ) b^4+A c f \left (15 c^2 d^2+46 a c f d+43 a^2 f^2\right ) b^3+2 B c \left (2 c^3 d^3+5 a c^2 f d^2+4 a^2 c f^2 d-11 a^3 f^3\right ) b^2-4 A c^2 \left (2 c^3 d^3+9 a c^2 f d^2+24 a^2 c f^2 d+17 a^3 f^3\right ) b+24 a^2 B c^2 f (c d+a f)^2+c \left (3 B d f^2 b^5-2 A f^2 (4 c d+3 a f) b^4-B f \left (17 c^2 d^2+10 a c f d-3 a^2 f^2\right ) b^3+2 A c f \left (15 c^2 d^2+22 a c f d+19 a^2 f^2\right ) b^2+4 B c \left (2 c^3 d^3+11 a c^2 f d^2+4 a^2 c f^2 d-5 a^3 f^3\right ) b-8 A c^2 (c d+a f)^2 (2 c d+5 a f)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c^2 d^2+2 a c f d-f \left (b^2 d-a^2 f\right )\right )^2 \sqrt{c x^2+b x+a}}-\frac{2 \left (a B \left (-f b^2+2 c^2 d+2 a c f\right )+A \left (b^3 f-b c (c d+3 a f)\right )+c \left (A f b^2+B (c d-a f) b-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (c x^2+b x+a\right )^{3/2}} \]

[Out]

(-2*(a*B*(2*c^2*d - b^2*f + 2*a*c*f) + A*(b^3*f - b*c*(c*d + 3*a*f)) + c*(A*b^2*
f + b*B*(c*d - a*f) - 2*A*c*(c*d + a*f))*x))/(3*(b^2 - 4*a*c)*(b^2*d*f - (c*d +
a*f)^2)*(a + b*x + c*x^2)^(3/2)) - (2*(3*b^6*B*d*f^2 + 24*a^2*B*c^2*f*(c*d + a*f
)^2 - A*b^5*f^2*(7*c*d + 6*a*f) - b^4*B*f*(7*c^2*d^2 + 14*a*c*d*f - 3*a^2*f^2) +
A*b^3*c*f*(15*c^2*d^2 + 46*a*c*d*f + 43*a^2*f^2) + 2*b^2*B*c*(2*c^3*d^3 + 5*a*c
^2*d^2*f + 4*a^2*c*d*f^2 - 11*a^3*f^3) - 4*A*b*c^2*(2*c^3*d^3 + 9*a*c^2*d^2*f +
24*a^2*c*d*f^2 + 17*a^3*f^3) + c*(3*b^5*B*d*f^2 - 2*A*b^4*f^2*(4*c*d + 3*a*f) -
8*A*c^2*(c*d + a*f)^2*(2*c*d + 5*a*f) - b^3*B*f*(17*c^2*d^2 + 10*a*c*d*f - 3*a^2
*f^2) + 2*A*b^2*c*f*(15*c^2*d^2 + 22*a*c*d*f + 19*a^2*f^2) + 4*b*B*c*(2*c^3*d^3
+ 11*a*c^2*d^2*f + 4*a^2*c*d*f^2 - 5*a^3*f^3))*x))/(3*(b^2 - 4*a*c)^2*(c^2*d^2 +
2*a*c*d*f - f*(b^2*d - a^2*f))^2*Sqrt[a + b*x + c*x^2]) - ((B*Sqrt[d] - A*Sqrt[
f])*f^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*S
qrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d - b*
Sqrt[d]*Sqrt[f] + a*f)^(5/2)) + ((B*Sqrt[d] + A*Sqrt[f])*f^(3/2)*ArcTanh[(b*Sqrt
[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f]
+ a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(5/2
))

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Rubi [A]  time = 4.31787, antiderivative size = 796, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\left (B \sqrt{d}-A \sqrt{f}\right ) \tanh ^{-1}\left (\frac{-2 \sqrt{f} a+\left (2 c \sqrt{d}-b \sqrt{f}\right ) x+b \sqrt{d}}{2 \sqrt{-\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (-\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}+\frac{\left (\sqrt{f} A+B \sqrt{d}\right ) \tanh ^{-1}\left (\frac{2 \sqrt{f} a+\left (\sqrt{f} b+2 c \sqrt{d}\right ) x+b \sqrt{d}}{2 \sqrt{\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}-\frac{2 \left (3 B d f^2 b^6-A f^2 (7 c d+6 a f) b^5-B f \left (7 c^2 d^2+14 a c f d-3 a^2 f^2\right ) b^4+A c f \left (15 c^2 d^2+46 a c f d+43 a^2 f^2\right ) b^3+2 B c \left (2 c^3 d^3+5 a c^2 f d^2+4 a^2 c f^2 d-11 a^3 f^3\right ) b^2-4 A c^2 \left (2 c^3 d^3+9 a c^2 f d^2+24 a^2 c f^2 d+17 a^3 f^3\right ) b+24 a^2 B c^2 f (c d+a f)^2+c \left (3 B d f^2 b^5-2 A f^2 (4 c d+3 a f) b^4-B f \left (17 c^2 d^2+10 a c f d-3 a^2 f^2\right ) b^3+2 A c f \left (15 c^2 d^2+22 a c f d+19 a^2 f^2\right ) b^2+4 B c \left (2 c^3 d^3+11 a c^2 f d^2+4 a^2 c f^2 d-5 a^3 f^3\right ) b-8 A c^2 (c d+a f)^2 (2 c d+5 a f)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c^2 d^2+2 a c f d-f \left (b^2 d-a^2 f\right )\right )^2 \sqrt{c x^2+b x+a}}-\frac{2 \left (A f b^3-A c (c d+3 a f) b+a B \left (-f b^2+2 c^2 d+2 a c f\right )+c \left (A f b^2+B (c d-a f) b-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (c x^2+b x+a\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x + c*x^2)^(5/2)*(d - f*x^2)),x]

[Out]

(-2*(A*b^3*f - A*b*c*(c*d + 3*a*f) + a*B*(2*c^2*d - b^2*f + 2*a*c*f) + c*(A*b^2*
f + b*B*(c*d - a*f) - 2*A*c*(c*d + a*f))*x))/(3*(b^2 - 4*a*c)*(b^2*d*f - (c*d +
a*f)^2)*(a + b*x + c*x^2)^(3/2)) - (2*(3*b^6*B*d*f^2 + 24*a^2*B*c^2*f*(c*d + a*f
)^2 - A*b^5*f^2*(7*c*d + 6*a*f) - b^4*B*f*(7*c^2*d^2 + 14*a*c*d*f - 3*a^2*f^2) +
A*b^3*c*f*(15*c^2*d^2 + 46*a*c*d*f + 43*a^2*f^2) + 2*b^2*B*c*(2*c^3*d^3 + 5*a*c
^2*d^2*f + 4*a^2*c*d*f^2 - 11*a^3*f^3) - 4*A*b*c^2*(2*c^3*d^3 + 9*a*c^2*d^2*f +
24*a^2*c*d*f^2 + 17*a^3*f^3) + c*(3*b^5*B*d*f^2 - 2*A*b^4*f^2*(4*c*d + 3*a*f) -
8*A*c^2*(c*d + a*f)^2*(2*c*d + 5*a*f) - b^3*B*f*(17*c^2*d^2 + 10*a*c*d*f - 3*a^2
*f^2) + 2*A*b^2*c*f*(15*c^2*d^2 + 22*a*c*d*f + 19*a^2*f^2) + 4*b*B*c*(2*c^3*d^3
+ 11*a*c^2*d^2*f + 4*a^2*c*d*f^2 - 5*a^3*f^3))*x))/(3*(b^2 - 4*a*c)^2*(c^2*d^2 +
2*a*c*d*f - f*(b^2*d - a^2*f))^2*Sqrt[a + b*x + c*x^2]) - ((B*Sqrt[d] - A*Sqrt[
f])*f^(3/2)*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*S
qrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d - b*
Sqrt[d]*Sqrt[f] + a*f)^(5/2)) + ((B*Sqrt[d] + A*Sqrt[f])*f^(3/2)*ArcTanh[(b*Sqrt
[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f]
+ a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(5/2
))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(c*x**2+b*x+a)**(5/2)/(-f*x**2+d),x)

[Out]

Timed out

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Mathematica [B]  time = 7.37118, size = 1847, normalized size = 2.32 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x + c*x^2)^(5/2)*(d - f*x^2)),x]

[Out]

((a + b*x + c*x^2)^3*((-2*(-(A*b*c^2*d) + 2*a*B*c^2*d + A*b^3*f - a*b^2*B*f - 3*
a*A*b*c*f + 2*a^2*B*c*f + b*B*c^2*d*x - 2*A*c^3*d*x + A*b^2*c*f*x - a*b*B*c*f*x
- 2*a*A*c^2*f*x))/(3*(b^2 - 4*a*c)*(-(c^2*d^2) + b^2*d*f - 2*a*c*d*f - a^2*f^2)*
(a + b*x + c*x^2)^2) - (2*(4*b^2*B*c^4*d^3 - 8*A*b*c^5*d^3 - 7*b^4*B*c^2*d^2*f +
15*A*b^3*c^3*d^2*f + 10*a*b^2*B*c^3*d^2*f - 36*a*A*b*c^4*d^2*f + 24*a^2*B*c^4*d
^2*f + 3*b^6*B*d*f^2 - 7*A*b^5*c*d*f^2 - 14*a*b^4*B*c*d*f^2 + 46*a*A*b^3*c^2*d*f
^2 + 8*a^2*b^2*B*c^2*d*f^2 - 96*a^2*A*b*c^3*d*f^2 + 48*a^3*B*c^3*d*f^2 - 6*a*A*b
^5*f^3 + 3*a^2*b^4*B*f^3 + 43*a^2*A*b^3*c*f^3 - 22*a^3*b^2*B*c*f^3 - 68*a^3*A*b*
c^2*f^3 + 24*a^4*B*c^2*f^3 + 8*b*B*c^5*d^3*x - 16*A*c^6*d^3*x - 17*b^3*B*c^3*d^2
*f*x + 30*A*b^2*c^4*d^2*f*x + 44*a*b*B*c^4*d^2*f*x - 72*a*A*c^5*d^2*f*x + 3*b^5*
B*c*d*f^2*x - 8*A*b^4*c^2*d*f^2*x - 10*a*b^3*B*c^2*d*f^2*x + 44*a*A*b^2*c^3*d*f^
2*x + 16*a^2*b*B*c^3*d*f^2*x - 96*a^2*A*c^4*d*f^2*x - 6*a*A*b^4*c*f^3*x + 3*a^2*
b^3*B*c*f^3*x + 38*a^2*A*b^2*c^2*f^3*x - 20*a^3*b*B*c^2*f^3*x - 40*a^3*A*c^3*f^3
*x))/(3*(b^2 - 4*a*c)^2*(-(c^2*d^2) + b^2*d*f - 2*a*c*d*f - a^2*f^2)^2*(a + b*x
+ c*x^2))))/(a + x*(b + c*x))^(5/2) - (f*(B*c^2*d^(5/2)*Sqrt[f] - 2*b*B*c*d^2*f
+ A*c^2*d^2*f + b^2*B*d^(3/2)*f^(3/2) - 2*A*b*c*d^(3/2)*f^(3/2) + 2*a*B*c*d^(3/2
)*f^(3/2) + A*b^2*d*f^2 - 2*a*b*B*d*f^2 + 2*a*A*c*d*f^2 - 2*a*A*b*Sqrt[d]*f^(5/2
) + a^2*B*Sqrt[d]*f^(5/2) + a^2*A*f^3)*(a + b*x + c*x^2)^(5/2)*Log[Sqrt[d]*Sqrt[
f] - f*x])/(2*Sqrt[d]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*(c^2*d^2 - b^2*d*f + 2
*a*c*d*f + a^2*f^2)^2*(a + x*(b + c*x))^(5/2)) + (f*(-(B*c^2*d^(5/2)*Sqrt[f]) -
2*b*B*c*d^2*f + A*c^2*d^2*f - b^2*B*d^(3/2)*f^(3/2) + 2*A*b*c*d^(3/2)*f^(3/2) -
2*a*B*c*d^(3/2)*f^(3/2) + A*b^2*d*f^2 - 2*a*b*B*d*f^2 + 2*a*A*c*d*f^2 + 2*a*A*b*
Sqrt[d]*f^(5/2) - a^2*B*Sqrt[d]*f^(5/2) + a^2*A*f^3)*(a + b*x + c*x^2)^(5/2)*Log
[Sqrt[d]*Sqrt[f] + f*x])/(2*Sqrt[d]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*(c^2*d^2
- b^2*d*f + 2*a*c*d*f + a^2*f^2)^2*(a + x*(b + c*x))^(5/2)) - (f*(-(B*c^2*d^(5/
2)*Sqrt[f]) - 2*b*B*c*d^2*f + A*c^2*d^2*f - b^2*B*d^(3/2)*f^(3/2) + 2*A*b*c*d^(3
/2)*f^(3/2) - 2*a*B*c*d^(3/2)*f^(3/2) + A*b^2*d*f^2 - 2*a*b*B*d*f^2 + 2*a*A*c*d*
f^2 + 2*a*A*b*Sqrt[d]*f^(5/2) - a^2*B*Sqrt[d]*f^(5/2) + a^2*A*f^3)*(a + b*x + c*
x^2)^(5/2)*Log[-(b*d) + 2*a*Sqrt[d]*Sqrt[f] - 2*c*d*x + b*Sqrt[d]*Sqrt[f]*x + 2*
Sqrt[d]*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2]])/(2*Sqrt[d]*S
qrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*(c^2*d^2 - b^2*d*f + 2*a*c*d*f + a^2*f^2)^2*(
a + x*(b + c*x))^(5/2)) + (f*(B*c^2*d^(5/2)*Sqrt[f] - 2*b*B*c*d^2*f + A*c^2*d^2*
f + b^2*B*d^(3/2)*f^(3/2) - 2*A*b*c*d^(3/2)*f^(3/2) + 2*a*B*c*d^(3/2)*f^(3/2) +
A*b^2*d*f^2 - 2*a*b*B*d*f^2 + 2*a*A*c*d*f^2 - 2*a*A*b*Sqrt[d]*f^(5/2) + a^2*B*Sq
rt[d]*f^(5/2) + a^2*A*f^3)*(a + b*x + c*x^2)^(5/2)*Log[b*d + 2*a*Sqrt[d]*Sqrt[f]
+ 2*c*d*x + b*Sqrt[d]*Sqrt[f]*x + 2*Sqrt[d]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]
*Sqrt[a + b*x + c*x^2]])/(2*Sqrt[d]*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*(c^2*d^2
- b^2*d*f + 2*a*c*d*f + a^2*f^2)^2*(a + x*(b + c*x))^(5/2))

_______________________________________________________________________________________

Maple [B]  time = 0.04, size = 6422, normalized size = 8.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(c*x^2+b*x+a)^(5/2)/(-f*x^2+d),x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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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((B*x+A)/(c*x**2+b*x+a)**(5/2)/(-f*x**2+d),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(B*x + A)/((c*x^2 + b*x + a)^(5/2)*(f*x^2 - d)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError