3.1166 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{d+e x} \, dx\)

Optimal. Leaf size=200 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (-b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{4 c^{3/2} e^3}-\frac{\sqrt{d} (B d-A e) \sqrt{c d-b e} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^3}-\frac{\sqrt{b x+c x^2} (-4 A c e-b B e+4 B c d-2 B c e x)}{4 c e^2} \]

[Out]

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

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Rubi [A]  time = 0.628787, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (-b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{4 c^{3/2} e^3}-\frac{\sqrt{d} (B d-A e) \sqrt{c d-b e} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^3}-\frac{\sqrt{b x+c x^2} (-4 A c e-b B e+4 B c d-2 B c e x)}{4 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 67.9328, size = 184, normalized size = 0.92 \[ - \frac{\sqrt{d} \left (A e - B d\right ) \sqrt{b e - c d} \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{e^{3}} + \frac{\sqrt{b x + c x^{2}} \left (2 A c e + B c e x + \frac{B \left (b e - 4 c d\right )}{2}\right )}{2 c e^{2}} - \frac{\left (- 4 A c e \left (b e - 2 c d\right ) + B \left (b^{2} e^{2} + 4 b c d e - 8 c^{2} d^{2}\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 c^{\frac{3}{2}} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.684815, size = 207, normalized size = 1.03 \[ \frac{\sqrt{x (b+c x)} \left (\frac{\log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (4 A c e (b e-2 c d)+B \left (-b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{c^{3/2} \sqrt{b+c x}}+\frac{e \sqrt{x} (4 A c e+b B e-4 B c d)}{c}+\frac{8 \sqrt{d} (B d-A e) \sqrt{b e-c d} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b+c x}}+2 B e^2 x^{3/2}\right )}{4 e^3 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 1069, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.05029, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(8*(B*c*d - A*c*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*(2*B*c*e^2*x - 4*B*c*d
*e + (B*b + 4*A*c)*e^2)*sqrt(c*x^2 + b*x)*sqrt(c) - (8*B*c^2*d^2 - 4*(B*b*c + 2*
A*c^2)*d*e - (B*b^2 - 4*A*b*c)*e^2)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x
)*c))/(c^(3/2)*e^3), -1/8*(16*(B*c*d - A*c*e)*sqrt(-c*d^2 + b*d*e)*sqrt(c)*arcta
n(sqrt(c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) - 2*(2*B*c*e^2*x - 4*B*c*d*e + (
B*b + 4*A*c)*e^2)*sqrt(c*x^2 + b*x)*sqrt(c) - (8*B*c^2*d^2 - 4*(B*b*c + 2*A*c^2)
*d*e - (B*b^2 - 4*A*b*c)*e^2)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/
(c^(3/2)*e^3), -1/4*(4*(B*c*d - A*c*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*B*c*e^
2*x - 4*B*c*d*e + (B*b + 4*A*c)*e^2)*sqrt(c*x^2 + b*x)*sqrt(-c) - (8*B*c^2*d^2 -
4*(B*b*c + 2*A*c^2)*d*e - (B*b^2 - 4*A*b*c)*e^2)*arctan(sqrt(c*x^2 + b*x)*sqrt(
-c)/(c*x)))/(sqrt(-c)*c*e^3), -1/4*(8*(B*c*d - A*c*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*B*c*e^2*x - 4*B*c*
d*e + (B*b + 4*A*c)*e^2)*sqrt(c*x^2 + b*x)*sqrt(-c) - (8*B*c^2*d^2 - 4*(B*b*c +
2*A*c^2)*d*e - (B*b^2 - 4*A*b*c)*e^2)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/
(sqrt(-c)*c*e^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x*(b + c*x))*(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(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d),x, algorithm="giac")

[Out]

Exception raised: TypeError