3.2464 \(\int \frac{(A+B x) (d+e x)^3}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=407 \[ \frac{\sqrt{a+b x+c x^2} \left (2 c e x \left (B \left (-4 c e (9 a e+16 b d)+35 b^2 e^2+24 c^2 d^2\right )+40 A c e (2 c d-b e)\right )+8 A c e \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+64 c^2 d^2\right )+B \left (-8 c^2 d e (48 a e+47 b d)+20 b c e^2 (11 a e+18 b d)-105 b^3 e^3+96 c^3 d^3\right )\right )}{192 c^4}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (24 b^2 c e \left (-5 a B e^2+6 A c d e+6 B c d^2\right )-32 b c^2 \left (-3 a A e^3-9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )-3 a B e \left (4 c d^2-a e^2\right )\right )-40 b^3 c e^2 (A e+3 B d)+35 b^4 B e^3\right )}{128 c^{9/2}}+\frac{(d+e x)^2 \sqrt{a+b x+c x^2} (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac{B (d+e x)^3 \sqrt{a+b x+c x^2}}{4 c} \]

[Out]

((6*B*c*d - 7*b*B*e + 8*A*c*e)*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(24*c^2) + (B*
(d + e*x)^3*Sqrt[a + b*x + c*x^2])/(4*c) + ((8*A*c*e*(64*c^2*d^2 + 15*b^2*e^2 -
2*c*e*(27*b*d + 8*a*e)) + B*(96*c^3*d^3 - 105*b^3*e^3 + 20*b*c*e^2*(18*b*d + 11*
a*e) - 8*c^2*d*e*(47*b*d + 48*a*e)) + 2*c*e*(40*A*c*e*(2*c*d - b*e) + B*(24*c^2*
d^2 + 35*b^2*e^2 - 4*c*e*(16*b*d + 9*a*e)))*x)*Sqrt[a + b*x + c*x^2])/(192*c^4)
+ ((35*b^4*B*e^3 - 40*b^3*c*e^2*(3*B*d + A*e) + 24*b^2*c*e*(6*B*c*d^2 + 6*A*c*d*
e - 5*a*B*e^2) - 32*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e - 9*a*B*d*e^2 - 3*a*A*e^3) +
16*c^2*(4*A*c*d*(2*c*d^2 - 3*a*e^2) - 3*a*B*e*(4*c*d^2 - a*e^2)))*ArcTanh[(b + 2
*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(9/2))

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Rubi [A]  time = 1.70019, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{\sqrt{a+b x+c x^2} \left (2 c e x \left (B \left (-4 c e (9 a e+16 b d)+35 b^2 e^2+24 c^2 d^2\right )+40 A c e (2 c d-b e)\right )+8 A c e \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+64 c^2 d^2\right )+B \left (-8 c^2 d e (48 a e+47 b d)+20 b c e^2 (11 a e+18 b d)-105 b^3 e^3+96 c^3 d^3\right )\right )}{192 c^4}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (24 b^2 c e \left (-5 a B e^2+6 A c d e+6 B c d^2\right )-32 b c^2 \left (-3 a A e^3-9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )-3 a B e \left (4 c d^2-a e^2\right )\right )-40 b^3 c e^2 (A e+3 B d)+35 b^4 B e^3\right )}{128 c^{9/2}}+\frac{(d+e x)^2 \sqrt{a+b x+c x^2} (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac{B (d+e x)^3 \sqrt{a+b x+c x^2}}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^3)/Sqrt[a + b*x + c*x^2],x]

[Out]

((6*B*c*d - 7*b*B*e + 8*A*c*e)*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(24*c^2) + (B*
(d + e*x)^3*Sqrt[a + b*x + c*x^2])/(4*c) + ((8*A*c*e*(64*c^2*d^2 + 15*b^2*e^2 -
2*c*e*(27*b*d + 8*a*e)) + B*(96*c^3*d^3 - 105*b^3*e^3 + 20*b*c*e^2*(18*b*d + 11*
a*e) - 8*c^2*d*e*(47*b*d + 48*a*e)) + 2*c*e*(40*A*c*e*(2*c*d - b*e) + B*(24*c^2*
d^2 + 35*b^2*e^2 - 4*c*e*(16*b*d + 9*a*e)))*x)*Sqrt[a + b*x + c*x^2])/(192*c^4)
+ ((35*b^4*B*e^3 - 40*b^3*c*e^2*(3*B*d + A*e) + 24*b^2*c*e*(6*B*c*d^2 + 6*A*c*d*
e - 5*a*B*e^2) - 32*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e - 9*a*B*d*e^2 - 3*a*A*e^3) +
16*c^2*(4*A*c*d*(2*c*d^2 - 3*a*e^2) - 3*a*B*e*(4*c*d^2 - a*e^2)))*ArcTanh[(b + 2
*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(9/2))

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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)*(e*x+d)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.771664, size = 355, normalized size = 0.87 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (B \left (-8 c^2 e \left (48 a d e+9 a e^2 x+54 b d^2+30 b d e x+7 b e^2 x^2\right )+10 b c e^2 (22 a e+36 b d+7 b e x)-105 b^3 e^3+48 c^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )-8 A c e \left (2 c e (8 a e+27 b d+5 b e x)-15 b^2 e^2-4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )\right )+3 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (24 b^2 c e \left (-5 a B e^2+6 A c d e+6 B c d^2\right )-32 b c^2 \left (-3 a A e^3-9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )+3 a B e \left (a e^2-4 c d^2\right )\right )-40 b^3 c e^2 (A e+3 B d)+35 b^4 B e^3\right )}{384 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^3)/Sqrt[a + b*x + c*x^2],x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-8*A*c*e*(-15*b^2*e^2 + 2*c*e*(27*b*d + 8*a*e
+ 5*b*e*x) - 4*c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + B*(-105*b^3*e^3 + 10*b*c*e^
2*(36*b*d + 22*a*e + 7*b*e*x) - 8*c^2*e*(54*b*d^2 + 48*a*d*e + 30*b*d*e*x + 9*a*
e^2*x + 7*b*e^2*x^2) + 48*c^3*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))) + 3*
(35*b^4*B*e^3 - 40*b^3*c*e^2*(3*B*d + A*e) + 24*b^2*c*e*(6*B*c*d^2 + 6*A*c*d*e -
5*a*B*e^2) - 32*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e - 9*a*B*d*e^2 - 3*a*A*e^3) + 16*
c^2*(4*A*c*d*(2*c*d^2 - 3*a*e^2) + 3*a*B*e*(-4*c*d^2 + a*e^2)))*Log[b + 2*c*x +
2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(384*c^(9/2))

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Maple [B]  time = 0.042, size = 981, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(1/2),x)

[Out]

9/4/c^(5/2)*b*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d*e^2-5/4/c^2*b*x*
(c*x^2+b*x+a)^(1/2)*B*d*e^2-9/4/c^2*b*(c*x^2+b*x+a)^(1/2)*A*d*e^2+3/2*x/c*(c*x^2
+b*x+a)^(1/2)*A*d*e^2+x^2/c*(c*x^2+b*x+a)^(1/2)*B*d*e^2-5/12/c^2*b*x*(c*x^2+b*x+
a)^(1/2)*A*e^3+15/8/c^3*b^2*(c*x^2+b*x+a)^(1/2)*B*d*e^2-15/16/c^(7/2)*b^3*ln((1/
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d*e^2+3/4/c^(5/2)*b*a*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))*A*e^3-2*a/c^2*(c*x^2+b*x+a)^(1/2)*B*d*e^2-7/24*B*e^3/c
^2*b*x^2*(c*x^2+b*x+a)^(1/2)+35/96*B*e^3/c^3*b^2*x*(c*x^2+b*x+a)^(1/2)-15/16*B*e
^3/c^(7/2)*b^2*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+55/48*B*e^3/c^3*b*a
*(c*x^2+b*x+a)^(1/2)-3/8*B*e^3*a/c^2*x*(c*x^2+b*x+a)^(1/2)+9/8/c^(5/2)*b^2*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d^2*e-3/2*a/c^(3/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))*A*d*e^2-3/2*a/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x
+a)^(1/2))*B*d^2*e-3/2*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*A*d
^2*e-9/4/c^2*b*(c*x^2+b*x+a)^(1/2)*B*d^2*e+9/8/c^(5/2)*b^2*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*A*d*e^2+3/2*x/c*(c*x^2+b*x+a)^(1/2)*B*d^2*e+1/c*(c*x^2+b*
x+a)^(1/2)*B*d^3+A*d^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+35/12
8*B*e^3/c^(9/2)*b^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/8*B*e^3*a^2/c^
(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/c*(c*x^2+b*x+a)^(1/2)*A*d^2*
e-1/2*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d^3+1/3*x^2/c*(c*x
^2+b*x+a)^(1/2)*A*e^3+5/8/c^3*b^2*(c*x^2+b*x+a)^(1/2)*A*e^3-5/16/c^(7/2)*b^3*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*A*e^3-2/3*a/c^2*(c*x^2+b*x+a)^(1/2)*A*e
^3-35/64*B*e^3/c^4*b^3*(c*x^2+b*x+a)^(1/2)+1/4*B*e^3*x^3/c*(c*x^2+b*x+a)^(1/2)

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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)*(e*x + d)^3/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.05747, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^3/sqrt(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

[1/768*(4*(48*B*c^3*e^3*x^3 + 192*B*c^3*d^3 - 144*(3*B*b*c^2 - 4*A*c^3)*d^2*e +
24*(15*B*b^2*c - 2*(8*B*a + 9*A*b)*c^2)*d*e^2 - (105*B*b^3 + 128*A*a*c^2 - 20*(1
1*B*a*b + 6*A*b^2)*c)*e^3 + 8*(24*B*c^3*d*e^2 - (7*B*b*c^2 - 8*A*c^3)*e^3)*x^2 +
2*(144*B*c^3*d^2*e - 24*(5*B*b*c^2 - 6*A*c^3)*d*e^2 + (35*B*b^2*c - 4*(9*B*a +
10*A*b)*c^2)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 3*(64*(B*b*c^3 - 2*A*c^4)*d
^3 - 48*(3*B*b^2*c^2 - 4*(B*a + A*b)*c^3)*d^2*e + 24*(5*B*b^3*c + 8*A*a*c^3 - 6*
(2*B*a*b + A*b^2)*c^2)*d*e^2 - (35*B*b^4 + 48*(B*a^2 + 2*A*a*b)*c^2 - 40*(3*B*a*
b^2 + A*b^3)*c)*e^3)*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 +
8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/c^(9/2), 1/384*(2*(48*B*c^3*e^3*x^3 + 192*B*c^
3*d^3 - 144*(3*B*b*c^2 - 4*A*c^3)*d^2*e + 24*(15*B*b^2*c - 2*(8*B*a + 9*A*b)*c^2
)*d*e^2 - (105*B*b^3 + 128*A*a*c^2 - 20*(11*B*a*b + 6*A*b^2)*c)*e^3 + 8*(24*B*c^
3*d*e^2 - (7*B*b*c^2 - 8*A*c^3)*e^3)*x^2 + 2*(144*B*c^3*d^2*e - 24*(5*B*b*c^2 -
6*A*c^3)*d*e^2 + (35*B*b^2*c - 4*(9*B*a + 10*A*b)*c^2)*e^3)*x)*sqrt(c*x^2 + b*x
+ a)*sqrt(-c) - 3*(64*(B*b*c^3 - 2*A*c^4)*d^3 - 48*(3*B*b^2*c^2 - 4*(B*a + A*b)*
c^3)*d^2*e + 24*(5*B*b^3*c + 8*A*a*c^3 - 6*(2*B*a*b + A*b^2)*c^2)*d*e^2 - (35*B*
b^4 + 48*(B*a^2 + 2*A*a*b)*c^2 - 40*(3*B*a*b^2 + A*b^3)*c)*e^3)*arctan(1/2*(2*c*
x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )^{3}}{\sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**3/sqrt(a + b*x + c*x**2), x)

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GIAC/XCAS [A]  time = 0.283318, size = 556, normalized size = 1.37 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (\frac{6 \, B x e^{3}}{c} + \frac{24 \, B c^{3} d e^{2} - 7 \, B b c^{2} e^{3} + 8 \, A c^{3} e^{3}}{c^{4}}\right )} x + \frac{144 \, B c^{3} d^{2} e - 120 \, B b c^{2} d e^{2} + 144 \, A c^{3} d e^{2} + 35 \, B b^{2} c e^{3} - 36 \, B a c^{2} e^{3} - 40 \, A b c^{2} e^{3}}{c^{4}}\right )} x + \frac{192 \, B c^{3} d^{3} - 432 \, B b c^{2} d^{2} e + 576 \, A c^{3} d^{2} e + 360 \, B b^{2} c d e^{2} - 384 \, B a c^{2} d e^{2} - 432 \, A b c^{2} d e^{2} - 105 \, B b^{3} e^{3} + 220 \, B a b c e^{3} + 120 \, A b^{2} c e^{3} - 128 \, A a c^{2} e^{3}}{c^{4}}\right )} + \frac{{\left (64 \, B b c^{3} d^{3} - 128 \, A c^{4} d^{3} - 144 \, B b^{2} c^{2} d^{2} e + 192 \, B a c^{3} d^{2} e + 192 \, A b c^{3} d^{2} e + 120 \, B b^{3} c d e^{2} - 288 \, B a b c^{2} d e^{2} - 144 \, A b^{2} c^{2} d e^{2} + 192 \, A a c^{3} d e^{2} - 35 \, B b^{4} e^{3} + 120 \, B a b^{2} c e^{3} + 40 \, A b^{3} c e^{3} - 48 \, B a^{2} c^{2} e^{3} - 96 \, A a b c^{2} e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^3/sqrt(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

1/192*sqrt(c*x^2 + b*x + a)*(2*(4*(6*B*x*e^3/c + (24*B*c^3*d*e^2 - 7*B*b*c^2*e^3
+ 8*A*c^3*e^3)/c^4)*x + (144*B*c^3*d^2*e - 120*B*b*c^2*d*e^2 + 144*A*c^3*d*e^2
+ 35*B*b^2*c*e^3 - 36*B*a*c^2*e^3 - 40*A*b*c^2*e^3)/c^4)*x + (192*B*c^3*d^3 - 43
2*B*b*c^2*d^2*e + 576*A*c^3*d^2*e + 360*B*b^2*c*d*e^2 - 384*B*a*c^2*d*e^2 - 432*
A*b*c^2*d*e^2 - 105*B*b^3*e^3 + 220*B*a*b*c*e^3 + 120*A*b^2*c*e^3 - 128*A*a*c^2*
e^3)/c^4) + 1/128*(64*B*b*c^3*d^3 - 128*A*c^4*d^3 - 144*B*b^2*c^2*d^2*e + 192*B*
a*c^3*d^2*e + 192*A*b*c^3*d^2*e + 120*B*b^3*c*d*e^2 - 288*B*a*b*c^2*d*e^2 - 144*
A*b^2*c^2*d*e^2 + 192*A*a*c^3*d*e^2 - 35*B*b^4*e^3 + 120*B*a*b^2*c*e^3 + 40*A*b^
3*c*e^3 - 48*B*a^2*c^2*e^3 - 96*A*a*b*c^2*e^3)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*sqrt(c) - b))/c^(9/2)