3.10 \(\int \frac{(a+b x) \sqrt{c+d x} (e+f x)}{x} \, dx\)

Optimal. Leaf size=77 \[ -\frac{2 (c+d x)^{3/2} (-5 d (a f+b e)+2 b c f-3 b d f x)}{15 d^2}+2 a e \sqrt{c+d x}-2 a \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right ) \]

[Out]

2*a*e*Sqrt[c + d*x] - (2*(c + d*x)^(3/2)*(2*b*c*f - 5*d*(b*e + a*f) - 3*b*d*f*x)
)/(15*d^2) - 2*a*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]]

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Rubi [A]  time = 0.114304, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{2 (c+d x)^{3/2} (-5 d (a f+b e)+2 b c f-3 b d f x)}{15 d^2}+2 a e \sqrt{c+d x}-2 a \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*Sqrt[c + d*x]*(e + f*x))/x,x]

[Out]

2*a*e*Sqrt[c + d*x] - (2*(c + d*x)^(3/2)*(2*b*c*f - 5*d*(b*e + a*f) - 3*b*d*f*x)
)/(15*d^2) - 2*a*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]]

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Rubi in Sympy [A]  time = 10.5927, size = 80, normalized size = 1.04 \[ - 2 a \sqrt{c} e \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )} + 2 a e \sqrt{c + d x} + \frac{4 \left (c + d x\right )^{\frac{3}{2}} \left (- b c f + \frac{3 b d f x}{2} + \frac{5 d \left (a f + b e\right )}{2}\right )}{15 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*sqrt(c)*e*atanh(sqrt(c + d*x)/sqrt(c)) + 2*a*e*sqrt(c + d*x) + 4*(c + d*x)*
*(3/2)*(-b*c*f + 3*b*d*f*x/2 + 5*d*(a*f + b*e)/2)/(15*d**2)

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Mathematica [A]  time = 0.214427, size = 81, normalized size = 1.05 \[ \frac{2 \sqrt{c+d x} (5 a d (c f+3 d e+d f x)-b (c+d x) (2 c f-5 d e-3 d f x))}{15 d^2}-2 a \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*Sqrt[c + d*x]*(e + f*x))/x,x]

[Out]

(2*Sqrt[c + d*x]*(-(b*(c + d*x)*(-5*d*e + 2*c*f - 3*d*f*x)) + 5*a*d*(3*d*e + c*f
+ d*f*x)))/(15*d^2) - 2*a*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]]

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Maple [A]  time = 0.017, size = 89, normalized size = 1.2 \[ 2\,{\frac{1}{{d}^{2}} \left ( 1/5\,fb \left ( dx+c \right ) ^{5/2}+1/3\, \left ( dx+c \right ) ^{3/2}adf-1/3\, \left ( dx+c \right ) ^{3/2}bcf+1/3\, \left ( dx+c \right ) ^{3/2}bde+a{d}^{2}e\sqrt{dx+c}-a\sqrt{c}{d}^{2}e{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/d^2*(1/5*f*b*(d*x+c)^(5/2)+1/3*(d*x+c)^(3/2)*a*d*f-1/3*(d*x+c)^(3/2)*b*c*f+1/3
*(d*x+c)^(3/2)*b*d*e+a*d^2*e*(d*x+c)^(1/2)-a*c^(1/2)*d^2*e*arctanh((d*x+c)^(1/2)
/c^(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)*sqrt(d*x + c)*(f*x + e)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242751, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a \sqrt{c} d^{2} e \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \,{\left (3 \, b d^{2} f x^{2} + 5 \,{\left (b c d + 3 \, a d^{2}\right )} e -{\left (2 \, b c^{2} - 5 \, a c d\right )} f +{\left (5 \, b d^{2} e +{\left (b c d + 5 \, a d^{2}\right )} f\right )} x\right )} \sqrt{d x + c}}{15 \, d^{2}}, -\frac{2 \,{\left (15 \, a \sqrt{-c} d^{2} e \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) -{\left (3 \, b d^{2} f x^{2} + 5 \,{\left (b c d + 3 \, a d^{2}\right )} e -{\left (2 \, b c^{2} - 5 \, a c d\right )} f +{\left (5 \, b d^{2} e +{\left (b c d + 5 \, a d^{2}\right )} f\right )} x\right )} \sqrt{d x + c}\right )}}{15 \, d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*sqrt(d*x + c)*(f*x + e)/x,x, algorithm="fricas")

[Out]

[1/15*(15*a*sqrt(c)*d^2*e*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(3*b*
d^2*f*x^2 + 5*(b*c*d + 3*a*d^2)*e - (2*b*c^2 - 5*a*c*d)*f + (5*b*d^2*e + (b*c*d
+ 5*a*d^2)*f)*x)*sqrt(d*x + c))/d^2, -2/15*(15*a*sqrt(-c)*d^2*e*arctan(sqrt(d*x
+ c)/sqrt(-c)) - (3*b*d^2*f*x^2 + 5*(b*c*d + 3*a*d^2)*e - (2*b*c^2 - 5*a*c*d)*f
+ (5*b*d^2*e + (b*c*d + 5*a*d^2)*f)*x)*sqrt(d*x + c))/d^2]

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Sympy [A]  time = 49.4732, size = 148, normalized size = 1.92 \[ - 2 a c e \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + d x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + d x \wedge - c < 0 \end{cases}\right ) + 2 a e \sqrt{c + d x} + \frac{2 b f \left (c + d x\right )^{\frac{5}{2}}}{5 d^{2}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (a d f - b c f + b d e\right )}{3 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*c*e*Piecewise((-atan(sqrt(c + d*x)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(sqrt
(c + d*x)/sqrt(c))/sqrt(c), (-c < 0) & (c < c + d*x)), (atanh(sqrt(c + d*x)/sqrt
(c))/sqrt(c), (-c < 0) & (c > c + d*x))) + 2*a*e*sqrt(c + d*x) + 2*b*f*(c + d*x)
**(5/2)/(5*d**2) + 2*(c + d*x)**(3/2)*(a*d*f - b*c*f + b*d*e)/(3*d**2)

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GIAC/XCAS [A]  time = 0.215528, size = 142, normalized size = 1.84 \[ \frac{2 \, a c \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) e}{\sqrt{-c}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} b d^{8} f - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} b c d^{8} f + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} a d^{9} f + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} b d^{9} e + 15 \, \sqrt{d x + c} a d^{10} e\right )}}{15 \, d^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*sqrt(d*x + c)*(f*x + e)/x,x, algorithm="giac")

[Out]

2*a*c*arctan(sqrt(d*x + c)/sqrt(-c))*e/sqrt(-c) + 2/15*(3*(d*x + c)^(5/2)*b*d^8*
f - 5*(d*x + c)^(3/2)*b*c*d^8*f + 5*(d*x + c)^(3/2)*a*d^9*f + 5*(d*x + c)^(3/2)*
b*d^9*e + 15*sqrt(d*x + c)*a*d^10*e)/d^10