3.1226 \(\int \frac{(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=263 \[ -\frac{2 (d+e x)^{5/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{5 e^6}+\frac{2 (d+e x)^{3/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac{2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt{d+e x}}-\frac{2 c (d+e x)^{7/2} (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac{2 d \sqrt{d+e x} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}+\frac{2 B c^2 (d+e x)^{9/2}}{9 e^6} \]

[Out]

(2*d^2*(B*d - A*e)*(c*d - b*e)^2)/(e^6*Sqrt[d + e*x]) + (2*d*(c*d - b*e)*(B*d*(5
*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e))*Sqrt[d + e*x])/e^6 + (2*(A*e*(6*c^2*d^2 - 6
*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*(d + e*x)^(3/2)
)/(3*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(d
+ e*x)^(5/2))/(5*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(7/2))/(7*e^
6) + (2*B*c^2*(d + e*x)^(9/2))/(9*e^6)

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Rubi [A]  time = 0.450453, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 (d+e x)^{5/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{5 e^6}+\frac{2 (d+e x)^{3/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac{2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt{d+e x}}-\frac{2 c (d+e x)^{7/2} (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac{2 d \sqrt{d+e x} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}+\frac{2 B c^2 (d+e x)^{9/2}}{9 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*d^2*(B*d - A*e)*(c*d - b*e)^2)/(e^6*Sqrt[d + e*x]) + (2*d*(c*d - b*e)*(B*d*(5
*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e))*Sqrt[d + e*x])/e^6 + (2*(A*e*(6*c^2*d^2 - 6
*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*(d + e*x)^(3/2)
)/(3*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(d
+ e*x)^(5/2))/(5*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(7/2))/(7*e^
6) + (2*B*c^2*(d + e*x)^(9/2))/(9*e^6)

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Rubi in Sympy [A]  time = 99.9382, size = 289, normalized size = 1.1 \[ \frac{2 B c^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{6}} + \frac{2 c \left (d + e x\right )^{\frac{7}{2}} \left (A c e + 2 B b e - 5 B c d\right )}{7 e^{6}} - \frac{2 d^{2} \left (A e - B d\right ) \left (b e - c d\right )^{2}}{e^{6} \sqrt{d + e x}} - \frac{2 d \sqrt{d + e x} \left (b e - c d\right ) \left (2 A b e^{2} - 4 A c d e - 3 B b d e + 5 B c d^{2}\right )}{e^{6}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{5 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{3 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*c**2*(d + e*x)**(9/2)/(9*e**6) + 2*c*(d + e*x)**(7/2)*(A*c*e + 2*B*b*e - 5*B
*c*d)/(7*e**6) - 2*d**2*(A*e - B*d)*(b*e - c*d)**2/(e**6*sqrt(d + e*x)) - 2*d*sq
rt(d + e*x)*(b*e - c*d)*(2*A*b*e**2 - 4*A*c*d*e - 3*B*b*d*e + 5*B*c*d**2)/e**6 +
2*(d + e*x)**(5/2)*(2*A*b*c*e**2 - 4*A*c**2*d*e + B*b**2*e**2 - 8*B*b*c*d*e + 1
0*B*c**2*d**2)/(5*e**6) + 2*(d + e*x)**(3/2)*(A*b**2*e**3 - 6*A*b*c*d*e**2 + 6*A
*c**2*d**2*e - 3*B*b**2*d*e**2 + 12*B*b*c*d**2*e - 10*B*c**2*d**3)/(3*e**6)

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Mathematica [A]  time = 0.363485, size = 273, normalized size = 1.04 \[ \frac{2 B \left (63 b^2 e^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+18 b c e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 c^2 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )-6 A e \left (35 b^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )-42 b c e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+3 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )}{315 e^6 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*A*e*(35*b^2*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) - 42*b*c*e*(16*d^3 + 8*d^2*e*x -
2*d*e^2*x^2 + e^3*x^3) + 3*c^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3
*x^3 - 5*e^4*x^4)) + 2*B*(63*b^2*e^2*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3
) + 18*b*c*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4)
+ 5*c^2*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4
+ 7*e^5*x^5)))/(315*e^6*Sqrt[d + e*x])

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Maple [A]  time = 0.011, size = 341, normalized size = 1.3 \[ -{\frac{-70\,B{c}^{2}{x}^{5}{e}^{5}-90\,A{c}^{2}{e}^{5}{x}^{4}-180\,Bbc{e}^{5}{x}^{4}+100\,B{c}^{2}d{e}^{4}{x}^{4}-252\,Abc{e}^{5}{x}^{3}+144\,A{c}^{2}d{e}^{4}{x}^{3}-126\,B{b}^{2}{e}^{5}{x}^{3}+288\,Bbcd{e}^{4}{x}^{3}-160\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}-210\,A{b}^{2}{e}^{5}{x}^{2}+504\,Abcd{e}^{4}{x}^{2}-288\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}+252\,B{b}^{2}d{e}^{4}{x}^{2}-576\,Bbc{d}^{2}{e}^{3}{x}^{2}+320\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}+840\,A{b}^{2}d{e}^{4}x-2016\,Abc{d}^{2}{e}^{3}x+1152\,A{c}^{2}{d}^{3}{e}^{2}x-1008\,B{b}^{2}{d}^{2}{e}^{3}x+2304\,Bbc{d}^{3}{e}^{2}x-1280\,B{c}^{2}{d}^{4}ex+1680\,A{b}^{2}{d}^{2}{e}^{3}-4032\,Abc{d}^{3}{e}^{2}+2304\,A{c}^{2}{d}^{4}e-2016\,B{b}^{2}{d}^{3}{e}^{2}+4608\,Bbc{d}^{4}e-2560\,B{c}^{2}{d}^{5}}{315\,{e}^{6}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(-35*B*c^2*e^5*x^5-45*A*c^2*e^5*x^4-90*B*b*c*e^5*x^4+50*B*c^2*d*e^4*x^4-1
26*A*b*c*e^5*x^3+72*A*c^2*d*e^4*x^3-63*B*b^2*e^5*x^3+144*B*b*c*d*e^4*x^3-80*B*c^
2*d^2*e^3*x^3-105*A*b^2*e^5*x^2+252*A*b*c*d*e^4*x^2-144*A*c^2*d^2*e^3*x^2+126*B*
b^2*d*e^4*x^2-288*B*b*c*d^2*e^3*x^2+160*B*c^2*d^3*e^2*x^2+420*A*b^2*d*e^4*x-1008
*A*b*c*d^2*e^3*x+576*A*c^2*d^3*e^2*x-504*B*b^2*d^2*e^3*x+1152*B*b*c*d^3*e^2*x-64
0*B*c^2*d^4*e*x+840*A*b^2*d^2*e^3-2016*A*b*c*d^3*e^2+1152*A*c^2*d^4*e-1008*B*b^2
*d^3*e^2+2304*B*b*c*d^4*e-1280*B*c^2*d^5)/(e*x+d)^(1/2)/e^6

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Maxima [A]  time = 0.700286, size = 404, normalized size = 1.54 \[ \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} B c^{2} - 45 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 315 \,{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} \sqrt{e x + d}}{e^{5}} + \frac{315 \,{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )}}{\sqrt{e x + d} e^{5}}\right )}}{315 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*((35*(e*x + d)^(9/2)*B*c^2 - 45*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d
)^(7/2) + 63*(10*B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*(e
*x + d)^(5/2) - 105*(10*B*c^2*d^3 - A*b^2*e^3 - 6*(2*B*b*c + A*c^2)*d^2*e + 3*(B
*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(3/2) + 315*(5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*(2
*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*sqrt(e*x + d))/e^5 + 315*(B
*c^2*d^5 - A*b^2*d^2*e^3 - (2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)/
(sqrt(e*x + d)*e^5))/e

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Fricas [A]  time = 0.272546, size = 390, normalized size = 1.48 \[ \frac{2 \,{\left (35 \, B c^{2} e^{5} x^{5} + 1280 \, B c^{2} d^{5} - 840 \, A b^{2} d^{2} e^{3} - 1152 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 1008 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 5 \,{\left (10 \, B c^{2} d e^{4} - 9 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} +{\left (80 \, B c^{2} d^{2} e^{3} - 72 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 63 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} -{\left (160 \, B c^{2} d^{3} e^{2} - 105 \, A b^{2} e^{5} - 144 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 126 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \,{\left (160 \, B c^{2} d^{4} e - 105 \, A b^{2} d e^{4} - 144 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 126 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )}}{315 \, \sqrt{e x + d} e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*c^2*e^5*x^5 + 1280*B*c^2*d^5 - 840*A*b^2*d^2*e^3 - 1152*(2*B*b*c + A
*c^2)*d^4*e + 1008*(B*b^2 + 2*A*b*c)*d^3*e^2 - 5*(10*B*c^2*d*e^4 - 9*(2*B*b*c +
A*c^2)*e^5)*x^4 + (80*B*c^2*d^2*e^3 - 72*(2*B*b*c + A*c^2)*d*e^4 + 63*(B*b^2 + 2
*A*b*c)*e^5)*x^3 - (160*B*c^2*d^3*e^2 - 105*A*b^2*e^5 - 144*(2*B*b*c + A*c^2)*d^
2*e^3 + 126*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 4*(160*B*c^2*d^4*e - 105*A*b^2*d*e^4
- 144*(2*B*b*c + A*c^2)*d^3*e^2 + 126*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)/(sqrt(e*x +
d)*e^6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28949, size = 595, normalized size = 2.26 \[ \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} B c^{2} e^{48} - 225 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{2} d e^{48} + 630 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{2} d^{2} e^{48} - 1050 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{2} d^{3} e^{48} + 1575 \, \sqrt{x e + d} B c^{2} d^{4} e^{48} + 90 \,{\left (x e + d\right )}^{\frac{7}{2}} B b c e^{49} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} A c^{2} e^{49} - 504 \,{\left (x e + d\right )}^{\frac{5}{2}} B b c d e^{49} - 252 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{2} d e^{49} + 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} B b c d^{2} e^{49} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} d^{2} e^{49} - 2520 \, \sqrt{x e + d} B b c d^{3} e^{49} - 1260 \, \sqrt{x e + d} A c^{2} d^{3} e^{49} + 63 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{2} e^{50} + 126 \,{\left (x e + d\right )}^{\frac{5}{2}} A b c e^{50} - 315 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e^{50} - 630 \,{\left (x e + d\right )}^{\frac{3}{2}} A b c d e^{50} + 945 \, \sqrt{x e + d} B b^{2} d^{2} e^{50} + 1890 \, \sqrt{x e + d} A b c d^{2} e^{50} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{51} - 630 \, \sqrt{x e + d} A b^{2} d e^{51}\right )} e^{\left (-54\right )} + \frac{2 \,{\left (B c^{2} d^{5} - 2 \, B b c d^{4} e - A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, A b c d^{3} e^{2} - A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*(x*e + d)^(9/2)*B*c^2*e^48 - 225*(x*e + d)^(7/2)*B*c^2*d*e^48 + 630*(x
*e + d)^(5/2)*B*c^2*d^2*e^48 - 1050*(x*e + d)^(3/2)*B*c^2*d^3*e^48 + 1575*sqrt(x
*e + d)*B*c^2*d^4*e^48 + 90*(x*e + d)^(7/2)*B*b*c*e^49 + 45*(x*e + d)^(7/2)*A*c^
2*e^49 - 504*(x*e + d)^(5/2)*B*b*c*d*e^49 - 252*(x*e + d)^(5/2)*A*c^2*d*e^49 + 1
260*(x*e + d)^(3/2)*B*b*c*d^2*e^49 + 630*(x*e + d)^(3/2)*A*c^2*d^2*e^49 - 2520*s
qrt(x*e + d)*B*b*c*d^3*e^49 - 1260*sqrt(x*e + d)*A*c^2*d^3*e^49 + 63*(x*e + d)^(
5/2)*B*b^2*e^50 + 126*(x*e + d)^(5/2)*A*b*c*e^50 - 315*(x*e + d)^(3/2)*B*b^2*d*e
^50 - 630*(x*e + d)^(3/2)*A*b*c*d*e^50 + 945*sqrt(x*e + d)*B*b^2*d^2*e^50 + 1890
*sqrt(x*e + d)*A*b*c*d^2*e^50 + 105*(x*e + d)^(3/2)*A*b^2*e^51 - 630*sqrt(x*e +
d)*A*b^2*d*e^51)*e^(-54) + 2*(B*c^2*d^5 - 2*B*b*c*d^4*e - A*c^2*d^4*e + B*b^2*d^
3*e^2 + 2*A*b*c*d^3*e^2 - A*b^2*d^2*e^3)*e^(-6)/sqrt(x*e + d)