∖int(A+Bx)∖left(bx+cx2∖right)3∕2 ________________________________________________d+ex∖,dx

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3.1175 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx\)

Optimal. Leaf size=392 \[ \frac{\sqrt{b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{64 c^2 e^4}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (8 A c e \left (b^3 e^3+6 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right )-B \left (3 b^4 e^4+8 b^3 c d e^3+48 b^2 c^2 d^2 e^2-192 b c^3 d^3 e+128 c^4 d^4\right )\right )}{64 c^{5/2} e^5}-\frac{d^{3/2} (B d-A e) (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^5}-\frac{\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2} \]

[Out]

((8*A*c*e*(8*c^2*d^2 - 10*b*c*d*e + b^2*e^2) - B*(64*c^3*d^3 - 80*b*c^2*d^2*e +

8*b^2*c*d*e^2 + 3*b^3*e^3) - 2*c*e*(8*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 8*b*

c*d*e - 3*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*c^2*e^4) - ((8*B*c*d - 3*b*B*e - 8

*A*c*e - 6*B*c*e*x)*(b*x + c*x^2)^(3/2))/(24*c*e^2) - ((8*A*c*e*(16*c^3*d^3 - 24

*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3) - B*(128*c^4*d^4 - 192*b*c^3*d^3*e + 48*

b^2*c^2*d^2*e^2 + 8*b^3*c*d*e^3 + 3*b^4*e^4))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x

^2]])/(64*c^(5/2)*e^5) - (d^(3/2)*(B*d - A*e)*(c*d - b*e)^(3/2)*ArcTanh[(b*d + (

2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e^5

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Rubi [A] time = 1.27928, antiderivative size = 392, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{64 c^2 e^4}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (8 A c e \left (b^3 e^3+6 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right )-B \left (3 b^4 e^4+8 b^3 c d e^3+48 b^2 c^2 d^2 e^2-192 b c^3 d^3 e+128 c^4 d^4\right )\right )}{64 c^{5/2} e^5}-\frac{d^{3/2} (B d-A e) (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^5}-\frac{\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2} \]

Antiderivative was successfully veriﬁed.

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

[Out]

((8*A*c*e*(8*c^2*d^2 - 10*b*c*d*e + b^2*e^2) - B*(64*c^3*d^3 - 80*b*c^2*d^2*e +

8*b^2*c*d*e^2 + 3*b^3*e^3) - 2*c*e*(8*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 8*b*

c*d*e - 3*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*c^2*e^4) - ((8*B*c*d - 3*b*B*e - 8

*A*c*e - 6*B*c*e*x)*(b*x + c*x^2)^(3/2))/(24*c*e^2) - ((8*A*c*e*(16*c^3*d^3 - 24

*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3) - B*(128*c^4*d^4 - 192*b*c^3*d^3*e + 48*

b^2*c^2*d^2*e^2 + 8*b^3*c*d*e^3 + 3*b^4*e^4))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x

^2]])/(64*c^(5/2)*e^5) - (d^(3/2)*(B*d - A*e)*(c*d - b*e)^(3/2)*ArcTanh[(b*d + (

2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e^5

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Rubi in Sympy [A] time = 148.58, size = 398, normalized size = 1.02 \[ \frac{d^{\frac{3}{2}} \left (A e - B d\right ) \left (b e - c d\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{e^{5}} + \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}} \left (4 A c e + 3 B c e x + \frac{B \left (3 b e - 8 c d\right )}{2}\right )}{12 c e^{2}} - \frac{\sqrt{b x + c x^{2}} \left (b c d e \left (8 A c e + 3 B b e - 8 B c d\right ) + \frac{c e x \left (- 8 A c e \left (b e - 2 c d\right ) + B \left (3 b^{2} e^{2} + 8 b c d e - 16 c^{2} d^{2}\right )\right )}{2} + \left (\frac{b e}{4} - c d\right ) \left (- 8 A c e \left (b e - 2 c d\right ) + B \left (3 b^{2} e^{2} + 8 b c d e - 16 c^{2} d^{2}\right )\right )\right )}{16 c^{2} e^{4}} + \frac{\left (- b c d e \left (b e - 2 c d\right ) \left (8 A c e + 3 B b e - 8 B c d\right ) + \left (- 2 A c e \left (b e - 2 c d\right ) + \frac{B \left (3 b^{2} e^{2} + 8 b c d e - 16 c^{2} d^{2}\right )}{4}\right ) \left (b^{2} e^{2} + 4 b c d e - 8 c^{2} d^{2}\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{16 c^{\frac{5}{2}} e^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

d**(3/2)*(A*e - B*d)*(b*e - c*d)**(3/2)*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)

*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)))/e**5 + (b*x + c*x**2)**(3/2)*(4*A*c*e + 3*

B*c*e*x + B*(3*b*e - 8*c*d)/2)/(12*c*e**2) - sqrt(b*x + c*x**2)*(b*c*d*e*(8*A*c*

e + 3*B*b*e - 8*B*c*d) + c*e*x*(-8*A*c*e*(b*e - 2*c*d) + B*(3*b**2*e**2 + 8*b*c*

d*e - 16*c**2*d**2))/2 + (b*e/4 - c*d)*(-8*A*c*e*(b*e - 2*c*d) + B*(3*b**2*e**2

+ 8*b*c*d*e - 16*c**2*d**2)))/(16*c**2*e**4) + (-b*c*d*e*(b*e - 2*c*d)*(8*A*c*e

+ 3*B*b*e - 8*B*c*d) + (-2*A*c*e*(b*e - 2*c*d) + B*(3*b**2*e**2 + 8*b*c*d*e - 16

*c**2*d**2)/4)*(b**2*e**2 + 4*b*c*d*e - 8*c**2*d**2))*atanh(sqrt(c)*x/sqrt(b*x +

c*x**2))/(16*c**(5/2)*e**5)

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Mathematica [A] time = 1.27049, size = 385, normalized size = 0.98 \[ \frac{(x (b+c x))^{3/2} \left (\frac{e \sqrt{x} \left (8 A c e \left (3 b^2 e^2+2 b c e (7 e x-15 d)+4 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+B \left (-9 b^3 e^3+6 b^2 c e^2 (e x-4 d)+8 b c^2 e \left (30 d^2-14 d e x+9 e^2 x^2\right )-16 c^3 \left (12 d^3-6 d^2 e x+4 d e^2 x^2-3 e^3 x^3\right )\right )\right )}{c^2 (b+c x)}+\frac{3 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (B \left (3 b^4 e^4+8 b^3 c d e^3+48 b^2 c^2 d^2 e^2-192 b c^3 d^3 e+128 c^4 d^4\right )-8 A c e \left (b^3 e^3+6 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right )\right )}{c^{5/2} (b+c x)^{3/2}}-\frac{384 d^{3/2} (B d-A e) (b e-c d)^{3/2} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b+c x)^{3/2}}\right )}{192 e^5 x^{3/2}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

((x*(b + c*x))^(3/2)*((e*Sqrt[x]*(8*A*c*e*(3*b^2*e^2 + 2*b*c*e*(-15*d + 7*e*x) +

4*c^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + B*(-9*b^3*e^3 + 6*b^2*c*e^2*(-4*d + e*x)

+ 8*b*c^2*e*(30*d^2 - 14*d*e*x + 9*e^2*x^2) - 16*c^3*(12*d^3 - 6*d^2*e*x + 4*d*

e^2*x^2 - 3*e^3*x^3))))/(c^2*(b + c*x)) - (384*d^(3/2)*(B*d - A*e)*(-(c*d) + b*e

)^(3/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(b + c*x)^

(3/2) + (3*(-8*A*c*e*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3) + B

*(128*c^4*d^4 - 192*b*c^3*d^3*e + 48*b^2*c^2*d^2*e^2 + 8*b^3*c*d*e^3 + 3*b^4*e^4

))*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(c^(5/2)*(b + c*x)^(3/2))))/(192*e^5*

x^(3/2))

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Maple [B] time = 0.014, size = 2334, normalized size = 6. \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/e^4*d^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c*B-5/4/e^2

*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*d*A+5/4/e^3*(c*(d/e

+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*d^2*B+1/e^3*d^2*(c*(d/e+x)^

2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c*A+1/4/e*(c*(d/e+x)^2+(b*e-2*c*d

)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*b*A-1/e^4*d^3*ln((1/2*(b*e-2*c*d)/e+c*(d/e+

x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*A

-3/64*B/e*b^3/c^2*(c*x^2+b*x)^(1/2)+3/128*B/e*b^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)

+(c*x^2+b*x)^(1/2))-1/16/e/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(

d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*b^3*A+1/8/e/c*(c*(d/e+x)^

2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^2*A-2/e^5*d^4/(-d*(b*e-c*d)/e^2

)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*

(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b*c*B+2/e^4*

d^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*

(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/

(d/e+x))*b*c*A+1/4*B/e*(c*x^2+b*x)^(3/2)*x+1/8*B/e/c*(c*x^2+b*x)^(3/2)*b+1/e^4*d

^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(

b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(

d/e+x))*b^2*B+3/8/e^3*d^2*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+

(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)*b^2*B+3/2/e^3*d^2*ln((1/2*

(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/

e^2)^(1/2))*c^(1/2)*b*A-1/e^5*d^4/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^

2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d

/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^2*A-1/2/e^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*

(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c*d*A+1/e^5*d^4*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x)

)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*B-1

/3/e^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*B*d+1/e^6*d^5/(

-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-

c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+

x))*c^2*B-3/32*B/e*b^2/c*(c*x^2+b*x)^(1/2)*x-3/8/e^2*d*ln((1/2*(b*e-2*c*d)/e+c*(

d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/

2)*b^2*A-3/2/e^4*d^3*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-

2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b*B-1/4/e^2*(c*(d/e+x)^2+(b*e-2

*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*b*B*d-1/8/e^2/c*(c*(d/e+x)^2+(b*e-2*c*d

)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^2*B*d+1/16/e^2/c^(3/2)*ln((1/2*(b*e-2*c*d)/

e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*

b^3*B*d-1/e^3*d^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*

(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d

)/e^2)^(1/2))/(d/e+x))*b^2*A+1/2/e^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c

*d)/e^2)^(1/2)*x*c*d^2*B+1/3/e*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^

2)^(3/2)*A

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

[Out]

Exception raised: ValueError

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Fricas [A] time = 35.4722, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/384*(384*(B*c^3*d^3 + A*b*c^2*d*e^2 - (B*b*c^2 + A*c^3)*d^2*e)*sqrt(c*d^2 -

b*d*e)*sqrt(c)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b

*x))/(e*x + d)) - 2*(48*B*c^3*e^4*x^3 - 192*B*c^3*d^3*e + 48*(5*B*b*c^2 + 4*A*c^

3)*d^2*e^2 - 24*(B*b^2*c + 10*A*b*c^2)*d*e^3 - 3*(3*B*b^3 - 8*A*b^2*c)*e^4 - 8*(

8*B*c^3*d*e^3 - (9*B*b*c^2 + 8*A*c^3)*e^4)*x^2 + 2*(48*B*c^3*d^2*e^2 - 8*(7*B*b*

c^2 + 6*A*c^3)*d*e^3 + (3*B*b^2*c + 56*A*b*c^2)*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(c

) + 3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^

3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b^3*c)*e^4)*log((2

*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/(c^(5/2)*e^5), -1/384*(768*(B*c^3*d^

3 + A*b*c^2*d*e^2 - (B*b*c^2 + A*c^3)*d^2*e)*sqrt(-c*d^2 + b*d*e)*sqrt(c)*arctan

(sqrt(c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) - 2*(48*B*c^3*e^4*x^3 - 192*B*c^3

*d^3*e + 48*(5*B*b*c^2 + 4*A*c^3)*d^2*e^2 - 24*(B*b^2*c + 10*A*b*c^2)*d*e^3 - 3*

(3*B*b^3 - 8*A*b^2*c)*e^4 - 8*(8*B*c^3*d*e^3 - (9*B*b*c^2 + 8*A*c^3)*e^4)*x^2 +

2*(48*B*c^3*d^2*e^2 - 8*(7*B*b*c^2 + 6*A*c^3)*d*e^3 + (3*B*b^2*c + 56*A*b*c^2)*e

^4)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d

^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e^3 + (3

*B*b^4 - 8*A*b^3*c)*e^4)*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/(c^(5

/2)*e^5), -1/192*(192*(B*c^3*d^3 + A*b*c^2*d*e^2 - (B*b*c^2 + A*c^3)*d^2*e)*sqrt

(c*d^2 - b*d*e)*sqrt(-c)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt

(c*x^2 + b*x))/(e*x + d)) - (48*B*c^3*e^4*x^3 - 192*B*c^3*d^3*e + 48*(5*B*b*c^2

+ 4*A*c^3)*d^2*e^2 - 24*(B*b^2*c + 10*A*b*c^2)*d*e^3 - 3*(3*B*b^3 - 8*A*b^2*c)*e

^4 - 8*(8*B*c^3*d*e^3 - (9*B*b*c^2 + 8*A*c^3)*e^4)*x^2 + 2*(48*B*c^3*d^2*e^2 - 8

*(7*B*b*c^2 + 6*A*c^3)*d*e^3 + (3*B*b^2*c + 56*A*b*c^2)*e^4)*x)*sqrt(c*x^2 + b*x

)*sqrt(-c) - 3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3*e + 48*(B*b^2*c^2 +

4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b^3*c)*e^

4)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^2*e^5), -1/192*(384*(B*

c^3*d^3 + A*b*c^2*d*e^2 - (B*b*c^2 + A*c^3)*d^2*e)*sqrt(-c*d^2 + b*d*e)*sqrt(-c)

*arctan(sqrt(c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) - (48*B*c^3*e^4*x^3 - 192*

B*c^3*d^3*e + 48*(5*B*b*c^2 + 4*A*c^3)*d^2*e^2 - 24*(B*b^2*c + 10*A*b*c^2)*d*e^3

- 3*(3*B*b^3 - 8*A*b^2*c)*e^4 - 8*(8*B*c^3*d*e^3 - (9*B*b*c^2 + 8*A*c^3)*e^4)*x

^2 + 2*(48*B*c^3*d^2*e^2 - 8*(7*B*b*c^2 + 6*A*c^3)*d*e^3 + (3*B*b^2*c + 56*A*b*c

^2)*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*

c^4)*d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e^

3 + (3*B*b^4 - 8*A*b^3*c)*e^4)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-

c)*c^2*e^5)]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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