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3.1176 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=323 \[ -\frac{\sqrt{b x+c x^2} \left (2 c e x (-6 A c e-b B e+8 B c d)+6 A c e (4 c d-3 b e)-B \left (b^2 e^2-28 b c d e+32 c^2 d^2\right )\right )}{8 c e^4}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 b c e (4 B d-3 A e) (2 c d-b e)-\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) (-6 A c e-b B e+8 B c d)\right )}{8 c^{3/2} e^5}+\frac{\sqrt{d} \sqrt{c d-b e} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^5}+\frac{\left (b x+c x^2\right )^{3/2} (-3 A e+4 B d+B e x)}{3 e^2 (d+e x)} \]

[Out]

-((6*A*c*e*(4*c*d - 3*b*e) - B*(32*c^2*d^2 - 28*b*c*d*e + b^2*e^2) + 2*c*e*(8*B*
c*d - b*B*e - 6*A*c*e)*x)*Sqrt[b*x + c*x^2])/(8*c*e^4) + ((4*B*d - 3*A*e + B*e*x
)*(b*x + c*x^2)^(3/2))/(3*e^2*(d + e*x)) + ((4*b*c*e*(4*B*d - 3*A*e)*(2*c*d - b*
e) - (8*B*c*d - b*B*e - 6*A*c*e)*(8*c^2*d^2 - 4*b*c*d*e - b^2*e^2))*ArcTanh[(Sqr
t[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(3/2)*e^5) + (Sqrt[d]*Sqrt[c*d - b*e]*(B*d*(8*c
*d - 5*b*e) - 3*A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sq
rt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^5)

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Rubi [A]  time = 0.98514, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{b x+c x^2} \left (2 c e x (-6 A c e-b B e+8 B c d)+6 A c e (4 c d-3 b e)-B \left (b^2 e^2-28 b c d e+32 c^2 d^2\right )\right )}{8 c e^4}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 b c e (4 B d-3 A e) (2 c d-b e)-\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) (-6 A c e-b B e+8 B c d)\right )}{8 c^{3/2} e^5}+\frac{\sqrt{d} \sqrt{c d-b e} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^5}+\frac{\left (b x+c x^2\right )^{3/2} (-3 A e+4 B d+B e x)}{3 e^2 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

-((6*A*c*e*(4*c*d - 3*b*e) - B*(32*c^2*d^2 - 28*b*c*d*e + b^2*e^2) + 2*c*e*(8*B*
c*d - b*B*e - 6*A*c*e)*x)*Sqrt[b*x + c*x^2])/(8*c*e^4) + ((4*B*d - 3*A*e + B*e*x
)*(b*x + c*x^2)^(3/2))/(3*e^2*(d + e*x)) + ((4*b*c*e*(4*B*d - 3*A*e)*(2*c*d - b*
e) - (8*B*c*d - b*B*e - 6*A*c*e)*(8*c^2*d^2 - 4*b*c*d*e - b^2*e^2))*ArcTanh[(Sqr
t[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(3/2)*e^5) + (Sqrt[d]*Sqrt[c*d - b*e]*(B*d*(8*c
*d - 5*b*e) - 3*A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sq
rt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^5)

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Rubi in Sympy [A]  time = 123.065, size = 321, normalized size = 0.99 \[ - \frac{\sqrt{d} \sqrt{b e - c d} \left (3 A b e^{2} - 6 A c d e - 5 B b d e + 8 B c d^{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{2 e^{5}} - \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}} \left (3 A e - 4 B d - B e x\right )}{3 e^{2} \left (d + e x\right )} + \frac{\sqrt{b x + c x^{2}} \left (2 b c e \left (3 A e - 4 B d\right ) + c e x \left (6 A c e + B b e - 8 B c d\right ) + \left (\frac{b e}{2} - 2 c d\right ) \left (6 A c e + B b e - 8 B c d\right )\right )}{4 c e^{4}} - \frac{\left (- 4 b c e \left (3 A e - 4 B d\right ) \left (b e - 2 c d\right ) + \left (b^{2} e^{2} + 4 b c d e - 8 c^{2} d^{2}\right ) \left (6 A c e + B b e - 8 B c d\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{3}{2}} e^{5}} \]

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)**2,x)

[Out]

-sqrt(d)*sqrt(b*e - c*d)*(3*A*b*e**2 - 6*A*c*d*e - 5*B*b*d*e + 8*B*c*d**2)*atan(
(-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)))/(2*e**5
) - (b*x + c*x**2)**(3/2)*(3*A*e - 4*B*d - B*e*x)/(3*e**2*(d + e*x)) + sqrt(b*x
+ c*x**2)*(2*b*c*e*(3*A*e - 4*B*d) + c*e*x*(6*A*c*e + B*b*e - 8*B*c*d) + (b*e/2
- 2*c*d)*(6*A*c*e + B*b*e - 8*B*c*d))/(4*c*e**4) - (-4*b*c*e*(3*A*e - 4*B*d)*(b*
e - 2*c*d) + (b**2*e**2 + 4*b*c*d*e - 8*c**2*d**2)*(6*A*c*e + B*b*e - 8*B*c*d))*
atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(8*c**(3/2)*e**5)

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Mathematica [A]  time = 1.45152, size = 354, normalized size = 1.1 \[ \frac{(x (b+c x))^{3/2} \left (\frac{e \sqrt{x} \left (6 A c e \left (b e (9 d+5 e x)-2 c \left (6 d^2+3 d e x-e^2 x^2\right )\right )+B \left (3 b^2 e^2 (d+e x)+2 b c e \left (-42 d^2-23 d e x+7 e^2 x^2\right )+8 c^2 \left (12 d^3+6 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )\right )}{c (b+c x) (d+e x)}-\frac{3 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (B \left (b^3 e^3+12 b^2 c d e^2-72 b c^2 d^2 e+64 c^3 d^3\right )-6 A c e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )\right )}{c^{3/2} (b+c x)^{3/2}}-\frac{24 \sqrt{d} \sqrt{b e-c d} (3 A e (b e-2 c d)+B d (8 c d-5 b e)) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b+c x)^{3/2}}\right )}{24 e^5 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(3/2)*((e*Sqrt[x]*(6*A*c*e*(b*e*(9*d + 5*e*x) - 2*c*(6*d^2 + 3*d*
e*x - e^2*x^2)) + B*(3*b^2*e^2*(d + e*x) + 2*b*c*e*(-42*d^2 - 23*d*e*x + 7*e^2*x
^2) + 8*c^2*(12*d^3 + 6*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3))))/(c*(b + c*x)*(d + e*
x)) - (24*Sqrt[d]*Sqrt[-(c*d) + b*e]*(B*d*(8*c*d - 5*b*e) + 3*A*e*(-2*c*d + b*e)
)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(b + c*x)^(3/2)
- (3*(-6*A*c*e*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2) + B*(64*c^3*d^3 - 72*b*c^2*d^2*
e + 12*b^2*c*d*e^2 + b^3*e^3))*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(c^(3/2)*
(b + c*x)^(3/2))))/(24*e^5*x^(3/2))

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Maple [B]  time = 0.019, size = 4283, normalized size = 13.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/e^4*d^3/(b*e-c*d)/(-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*c*B-15/2/e^5*d^4/(b*e-c*d)/(-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^2*B+15/2/e^4*d^
3/(b*e-c*d)/(-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^2*A-6/e^3*d^2/(b*e-c*d)/(-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*c*A+B/e^4*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+1/3*B/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)+1/e/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/
e+x)-d*(b*e-c*d)/e^2)^(3/2)*b*B+1/d/(b*e-c*d)/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e
*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)*A-27/8/e^2*d/(b*e-c*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+6/e^3*d^2/(b*e-c*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^(3/2)*b*A-3/2/e^2*d/(b*e-c*d)
*(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^2*A-21/4/e^2*d/(b
*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*c*A-3/2/e^3*
d^2/(b*e-c*d)/(-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^3*B+3/2/e^2*d/(b*e-c*d)/(-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^3*A+3/e^6*d^5/(b*e-c*d)/(-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^3*B+2*B/e^5*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-1/2*B/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-3/8*B/e^3*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+3/2*B/e^4*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-B/e^4*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-B/e^6*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-c/d/(b*e-c*d
)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x*A+1/e*c/(b*e-c*d)*
(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x*B+3/8/e/(b*e-c*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^3*A-3/e^4*d^3/(b*e-c*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^(5/2)*
A+3/e^5*d^4/(b*e-c*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^(5/2)*B-1/e^2/(b*e-c*d)*(c*(d/e+x)^2
+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*B*d-9/4/e^2/(b*e-c*d)*(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+3/e^3*d^2/(b*e-c*d)*(c*(
d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2*A-3/e^4*d^3/(b*e-c*d)*
(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2*B-6/e^4*d^3/(b*e-c
*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^(3/2)*b*B+3/2/e/(b*e-c*d)*(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*c*A+27/8/e^3*d^2/(b*e-c*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*B+3/2/e^3*d^2/(b*e-c*d)*(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^2*B+21/4/e^3*d^2/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)
-d*(b*e-c*d)/e^2)^(1/2)*b*c*B-3/8/e^2/(b*e-c*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^3*
B*d-3/e^5*d^4/(b*e-c*d)/(-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^3*A-1/16*B/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^3-1/
d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b*A+1/e/(b
*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*A+9/4/e/(b*e
-c*d)*(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-1/e/(b*e-c
*d)/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)*B-B/e^5*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)-3/2/e^2/(b*e-c*d)*(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*c*B*d+1/4*B/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+1/8*B/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^2-5/4*B/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

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.81704, 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)^2,x, algorithm="fricas")

[Out]

[1/48*(24*(8*B*c^2*d^3 + 3*A*b*c*d*e^2 - (5*B*b*c + 6*A*c^2)*d^2*e + (8*B*c^2*d^
2*e + 3*A*b*c*e^3 - (5*B*b*c + 6*A*c^2)*d*e^2)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c)*lo
g((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*(8*B*c^2*e^4*x^3 + 96*B*c^2*d^3*e - 12*(7*B*b*c + 6*A*c^2)*d^2*e^2 + 3*(B*b^2
 + 18*A*b*c)*d*e^3 - 2*(8*B*c^2*d*e^3 - (7*B*b*c + 6*A*c^2)*e^4)*x^2 + (48*B*c^2
*d^2*e^2 - 2*(23*B*b*c + 18*A*c^2)*d*e^3 + 3*(B*b^2 + 10*A*b*c)*e^4)*x)*sqrt(c*x
^2 + b*x)*sqrt(c) - 3*(64*B*c^3*d^4 - 24*(3*B*b*c^2 + 2*A*c^3)*d^3*e + 12*(B*b^2
*c + 4*A*b*c^2)*d^2*e^2 + (B*b^3 - 6*A*b^2*c)*d*e^3 + (64*B*c^3*d^3*e - 24*(3*B*
b*c^2 + 2*A*c^3)*d^2*e^2 + 12*(B*b^2*c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A*b^2*c)*
e^4)*x)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/((c*e^6*x + c*d*e^5)*s
qrt(c)), 1/48*(48*(8*B*c^2*d^3 + 3*A*b*c*d*e^2 - (5*B*b*c + 6*A*c^2)*d^2*e + (8*
B*c^2*d^2*e + 3*A*b*c*e^3 - (5*B*b*c + 6*A*c^2)*d*e^2)*x)*sqrt(-c*d^2 + b*d*e)*s
qrt(c)*arctan(sqrt(c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) + 2*(8*B*c^2*e^4*x^3
 + 96*B*c^2*d^3*e - 12*(7*B*b*c + 6*A*c^2)*d^2*e^2 + 3*(B*b^2 + 18*A*b*c)*d*e^3
- 2*(8*B*c^2*d*e^3 - (7*B*b*c + 6*A*c^2)*e^4)*x^2 + (48*B*c^2*d^2*e^2 - 2*(23*B*
b*c + 18*A*c^2)*d*e^3 + 3*(B*b^2 + 10*A*b*c)*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(c) -
 3*(64*B*c^3*d^4 - 24*(3*B*b*c^2 + 2*A*c^3)*d^3*e + 12*(B*b^2*c + 4*A*b*c^2)*d^2
*e^2 + (B*b^3 - 6*A*b^2*c)*d*e^3 + (64*B*c^3*d^3*e - 24*(3*B*b*c^2 + 2*A*c^3)*d^
2*e^2 + 12*(B*b^2*c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A*b^2*c)*e^4)*x)*log((2*c*x
+ b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/((c*e^6*x + c*d*e^5)*sqrt(c)), 1/24*(12*(
8*B*c^2*d^3 + 3*A*b*c*d*e^2 - (5*B*b*c + 6*A*c^2)*d^2*e + (8*B*c^2*d^2*e + 3*A*b
*c*e^3 - (5*B*b*c + 6*A*c^2)*d*e^2)*x)*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)) + (8*B*c^2*
e^4*x^3 + 96*B*c^2*d^3*e - 12*(7*B*b*c + 6*A*c^2)*d^2*e^2 + 3*(B*b^2 + 18*A*b*c)
*d*e^3 - 2*(8*B*c^2*d*e^3 - (7*B*b*c + 6*A*c^2)*e^4)*x^2 + (48*B*c^2*d^2*e^2 - 2
*(23*B*b*c + 18*A*c^2)*d*e^3 + 3*(B*b^2 + 10*A*b*c)*e^4)*x)*sqrt(c*x^2 + b*x)*sq
rt(-c) - 3*(64*B*c^3*d^4 - 24*(3*B*b*c^2 + 2*A*c^3)*d^3*e + 12*(B*b^2*c + 4*A*b*
c^2)*d^2*e^2 + (B*b^3 - 6*A*b^2*c)*d*e^3 + (64*B*c^3*d^3*e - 24*(3*B*b*c^2 + 2*A
*c^3)*d^2*e^2 + 12*(B*b^2*c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A*b^2*c)*e^4)*x)*arc
tan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/((c*e^6*x + c*d*e^5)*sqrt(-c)), 1/24*(24*
(8*B*c^2*d^3 + 3*A*b*c*d*e^2 - (5*B*b*c + 6*A*c^2)*d^2*e + (8*B*c^2*d^2*e + 3*A*
b*c*e^3 - (5*B*b*c + 6*A*c^2)*d*e^2)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(-c)*arctan(sqr
t(c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) + (8*B*c^2*e^4*x^3 + 96*B*c^2*d^3*e -
 12*(7*B*b*c + 6*A*c^2)*d^2*e^2 + 3*(B*b^2 + 18*A*b*c)*d*e^3 - 2*(8*B*c^2*d*e^3
- (7*B*b*c + 6*A*c^2)*e^4)*x^2 + (48*B*c^2*d^2*e^2 - 2*(23*B*b*c + 18*A*c^2)*d*e
^3 + 3*(B*b^2 + 10*A*b*c)*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 3*(64*B*c^3*d^4 -
 24*(3*B*b*c^2 + 2*A*c^3)*d^3*e + 12*(B*b^2*c + 4*A*b*c^2)*d^2*e^2 + (B*b^3 - 6*
A*b^2*c)*d*e^3 + (64*B*c^3*d^3*e - 24*(3*B*b*c^2 + 2*A*c^3)*d^2*e^2 + 12*(B*b^2*
c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A*b^2*c)*e^4)*x)*arctan(sqrt(c*x^2 + b*x)*sqrt
(-c)/(c*x)))/((c*e^6*x + c*d*e^5)*sqrt(-c))]

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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)**(3/2)/(e*x+d)**2,x)

[Out]

Timed out

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

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)^2,x, algorithm="giac")

[Out]

Timed out

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