3.16 \(\int \frac{\sqrt{a+b x} (c+d x)^2 (e+f x)}{x} \, dx\)

Optimal. Leaf size=145 \[ \frac{2 (a+b x)^{3/2} \left (2 \left (4 a^2 d^2 f-7 a b d (2 c f+d e)+5 b^2 c (2 c f+7 d e)\right )+3 b d x (-4 a d f+4 b c f+7 b d e)\right )}{105 b^3}+2 c^2 e \sqrt{a+b x}-2 \sqrt{a} c^2 e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2 f (a+b x)^{3/2} (c+d x)^2}{7 b} \]

[Out]

2*c^2*e*Sqrt[a + b*x] + (2*f*(a + b*x)^(3/2)*(c + d*x)^2)/(7*b) + (2*(a + b*x)^(
3/2)*(2*(4*a^2*d^2*f - 7*a*b*d*(d*e + 2*c*f) + 5*b^2*c*(7*d*e + 2*c*f)) + 3*b*d*
(7*b*d*e + 4*b*c*f - 4*a*d*f)*x))/(105*b^3) - 2*Sqrt[a]*c^2*e*ArcTanh[Sqrt[a + b
*x]/Sqrt[a]]

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Rubi [A]  time = 0.319054, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 (a+b x)^{3/2} \left (2 \left (4 a^2 d^2 f-7 a b d (2 c f+d e)+5 b^2 c (2 c f+7 d e)\right )+3 b d x (-4 a d f+4 b c f+7 b d e)\right )}{105 b^3}+2 c^2 e \sqrt{a+b x}-2 \sqrt{a} c^2 e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2 f (a+b x)^{3/2} (c+d x)^2}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

2*c^2*e*Sqrt[a + b*x] + (2*f*(a + b*x)^(3/2)*(c + d*x)^2)/(7*b) + (2*(a + b*x)^(
3/2)*(2*(4*a^2*d^2*f - 7*a*b*d*(d*e + 2*c*f) + 5*b^2*c*(7*d*e + 2*c*f)) + 3*b*d*
(7*b*d*e + 4*b*c*f - 4*a*d*f)*x))/(105*b^3) - 2*Sqrt[a]*c^2*e*ArcTanh[Sqrt[a + b
*x]/Sqrt[a]]

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Rubi in Sympy [A]  time = 26.2438, size = 160, normalized size = 1.1 \[ - 2 \sqrt{a} c^{2} e \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 2 c^{2} e \sqrt{a + b x} + \frac{2 f \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{2}}{7 b} + \frac{8 \left (a + b x\right )^{\frac{3}{2}} \left (\frac{a d \left (4 a d f - 4 b c f - 7 b d e\right )}{2} - \frac{5 b c \left (2 a d f - 2 b c f - 7 b d e\right )}{2} - \frac{3 b d x \left (4 a d f - 4 b c f - 7 b d e\right )}{4}\right )}{105 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a)*c**2*e*atanh(sqrt(a + b*x)/sqrt(a)) + 2*c**2*e*sqrt(a + b*x) + 2*f*(a
+ b*x)**(3/2)*(c + d*x)**2/(7*b) + 8*(a + b*x)**(3/2)*(a*d*(4*a*d*f - 4*b*c*f -
7*b*d*e)/2 - 5*b*c*(2*a*d*f - 2*b*c*f - 7*b*d*e)/2 - 3*b*d*x*(4*a*d*f - 4*b*c*f
- 7*b*d*e)/4)/(105*b**3)

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Mathematica [A]  time = 0.35229, size = 157, normalized size = 1.08 \[ \frac{2 \sqrt{a+b x} \left (8 a^3 d^2 f-2 a^2 b d (14 c f+7 d e+2 d f x)+a b^2 \left (35 c^2 f+14 c d (5 e+f x)+d^2 x (7 e+3 f x)\right )+b^3 \left (35 c^2 (3 e+f x)+14 c d x (5 e+3 f x)+3 d^2 x^2 (7 e+5 f x)\right )\right )}{105 b^3}-2 \sqrt{a} c^2 e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x]*(8*a^3*d^2*f - 2*a^2*b*d*(7*d*e + 14*c*f + 2*d*f*x) + a*b^2*(35
*c^2*f + 14*c*d*(5*e + f*x) + d^2*x*(7*e + 3*f*x)) + b^3*(35*c^2*(3*e + f*x) + 1
4*c*d*x*(5*e + 3*f*x) + 3*d^2*x^2*(7*e + 5*f*x))))/(105*b^3) - 2*Sqrt[a]*c^2*e*A
rcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Maple [A]  time = 0.014, size = 176, normalized size = 1.2 \[ 2\,{\frac{1}{{b}^{3}} \left ( 1/7\,f{d}^{2} \left ( bx+a \right ) ^{7/2}-2/5\, \left ( bx+a \right ) ^{5/2}a{d}^{2}f+2/5\, \left ( bx+a \right ) ^{5/2}bcdf+1/5\, \left ( bx+a \right ) ^{5/2}b{d}^{2}e+1/3\, \left ( bx+a \right ) ^{3/2}{a}^{2}{d}^{2}f-2/3\, \left ( bx+a \right ) ^{3/2}abcdf-1/3\, \left ( bx+a \right ) ^{3/2}ab{d}^{2}e+1/3\, \left ( bx+a \right ) ^{3/2}{b}^{2}{c}^{2}f+2/3\, \left ( bx+a \right ) ^{3/2}{b}^{2}cde+{b}^{3}{c}^{2}e\sqrt{bx+a}-\sqrt{a}{b}^{3}{c}^{2}e{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/b^3*(1/7*f*d^2*(b*x+a)^(7/2)-2/5*(b*x+a)^(5/2)*a*d^2*f+2/5*(b*x+a)^(5/2)*b*c*d
*f+1/5*(b*x+a)^(5/2)*b*d^2*e+1/3*(b*x+a)^(3/2)*a^2*d^2*f-2/3*(b*x+a)^(3/2)*a*b*c
*d*f-1/3*(b*x+a)^(3/2)*a*b*d^2*e+1/3*(b*x+a)^(3/2)*b^2*c^2*f+2/3*(b*x+a)^(3/2)*b
^2*c*d*e+b^3*c^2*e*(b*x+a)^(1/2)-a^(1/2)*b^3*c^2*e*arctanh((b*x+a)^(1/2)/a^(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(sqrt(b*x + a)*(d*x + c)^2*(f*x + e)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229226, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, \sqrt{a} b^{3} c^{2} e \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (15 \, b^{3} d^{2} f x^{3} + 3 \,{\left (7 \, b^{3} d^{2} e +{\left (14 \, b^{3} c d + a b^{2} d^{2}\right )} f\right )} x^{2} + 7 \,{\left (15 \, b^{3} c^{2} + 10 \, a b^{2} c d - 2 \, a^{2} b d^{2}\right )} e +{\left (35 \, a b^{2} c^{2} - 28 \, a^{2} b c d + 8 \, a^{3} d^{2}\right )} f +{\left (7 \,{\left (10 \, b^{3} c d + a b^{2} d^{2}\right )} e +{\left (35 \, b^{3} c^{2} + 14 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} f\right )} x\right )} \sqrt{b x + a}}{105 \, b^{3}}, -\frac{2 \,{\left (105 \, \sqrt{-a} b^{3} c^{2} e \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (15 \, b^{3} d^{2} f x^{3} + 3 \,{\left (7 \, b^{3} d^{2} e +{\left (14 \, b^{3} c d + a b^{2} d^{2}\right )} f\right )} x^{2} + 7 \,{\left (15 \, b^{3} c^{2} + 10 \, a b^{2} c d - 2 \, a^{2} b d^{2}\right )} e +{\left (35 \, a b^{2} c^{2} - 28 \, a^{2} b c d + 8 \, a^{3} d^{2}\right )} f +{\left (7 \,{\left (10 \, b^{3} c d + a b^{2} d^{2}\right )} e +{\left (35 \, b^{3} c^{2} + 14 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} f\right )} x\right )} \sqrt{b x + a}\right )}}{105 \, b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/105*(105*sqrt(a)*b^3*c^2*e*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(
15*b^3*d^2*f*x^3 + 3*(7*b^3*d^2*e + (14*b^3*c*d + a*b^2*d^2)*f)*x^2 + 7*(15*b^3*
c^2 + 10*a*b^2*c*d - 2*a^2*b*d^2)*e + (35*a*b^2*c^2 - 28*a^2*b*c*d + 8*a^3*d^2)*
f + (7*(10*b^3*c*d + a*b^2*d^2)*e + (35*b^3*c^2 + 14*a*b^2*c*d - 4*a^2*b*d^2)*f)
*x)*sqrt(b*x + a))/b^3, -2/105*(105*sqrt(-a)*b^3*c^2*e*arctan(sqrt(b*x + a)/sqrt
(-a)) - (15*b^3*d^2*f*x^3 + 3*(7*b^3*d^2*e + (14*b^3*c*d + a*b^2*d^2)*f)*x^2 + 7
*(15*b^3*c^2 + 10*a*b^2*c*d - 2*a^2*b*d^2)*e + (35*a*b^2*c^2 - 28*a^2*b*c*d + 8*
a^3*d^2)*f + (7*(10*b^3*c*d + a*b^2*d^2)*e + (35*b^3*c^2 + 14*a*b^2*c*d - 4*a^2*
b*d^2)*f)*x)*sqrt(b*x + a))/b^3]

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Sympy [A]  time = 52.0002, size = 223, normalized size = 1.54 \[ - 2 a c^{2} e \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) + 2 c^{2} e \sqrt{a + b x} + \frac{2 d^{2} f \left (a + b x\right )^{\frac{7}{2}}}{7 b^{3}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (- 2 a d^{2} f + 2 b c d f + b d^{2} e\right )}{5 b^{3}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (a^{2} d^{2} f - 2 a b c d f - a b d^{2} e + b^{2} c^{2} f + 2 b^{2} c d e\right )}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*c**2*e*Piecewise((-atan(sqrt(a + b*x)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(s
qrt(a + b*x)/sqrt(a))/sqrt(a), (-a < 0) & (a < a + b*x)), (atanh(sqrt(a + b*x)/s
qrt(a))/sqrt(a), (-a < 0) & (a > a + b*x))) + 2*c**2*e*sqrt(a + b*x) + 2*d**2*f*
(a + b*x)**(7/2)/(7*b**3) + 2*(a + b*x)**(5/2)*(-2*a*d**2*f + 2*b*c*d*f + b*d**2
*e)/(5*b**3) + 2*(a + b*x)**(3/2)*(a**2*d**2*f - 2*a*b*c*d*f - a*b*d**2*e + b**2
*c**2*f + 2*b**2*c*d*e)/(3*b**3)

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GIAC/XCAS [A]  time = 0.235664, size = 271, normalized size = 1.87 \[ \frac{2 \, a c^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) e}{\sqrt{-a}} + \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{20} c^{2} f + 42 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{19} c d f - 70 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{19} c d f + 15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{18} d^{2} f - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{18} d^{2} f + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{18} d^{2} f + 105 \, \sqrt{b x + a} b^{21} c^{2} e + 70 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{20} c d e + 21 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{19} d^{2} e - 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{19} d^{2} e\right )}}{105 \, b^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*(d*x + c)^2*(f*x + e)/x,x, algorithm="giac")

[Out]

2*a*c^2*arctan(sqrt(b*x + a)/sqrt(-a))*e/sqrt(-a) + 2/105*(35*(b*x + a)^(3/2)*b^
20*c^2*f + 42*(b*x + a)^(5/2)*b^19*c*d*f - 70*(b*x + a)^(3/2)*a*b^19*c*d*f + 15*
(b*x + a)^(7/2)*b^18*d^2*f - 42*(b*x + a)^(5/2)*a*b^18*d^2*f + 35*(b*x + a)^(3/2
)*a^2*b^18*d^2*f + 105*sqrt(b*x + a)*b^21*c^2*e + 70*(b*x + a)^(3/2)*b^20*c*d*e
+ 21*(b*x + a)^(5/2)*b^19*d^2*e - 35*(b*x + a)^(3/2)*a*b^19*d^2*e)/b^21