3.8 \(\int \frac{(a+b x)^3 \sqrt{c+d x} (e+f x)}{x} \, dx\)

Optimal. Leaf size=227 \[ 2 a^3 e \sqrt{c+d x}-2 a^3 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+\frac{2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )\right )}{315 d^4}+\frac{2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{21 d^2}+\frac{2 f (a+b x)^3 (c+d x)^{3/2}}{9 d} \]

[Out]

2*a^3*e*Sqrt[c + d*x] + (2*(3*b*d*e - 2*b*c*f + 2*a*d*f)*(a + b*x)^2*(c + d*x)^(
3/2))/(21*d^2) + (2*f*(a + b*x)^3*(c + d*x)^(3/2))/(9*d) + (2*(c + d*x)^(3/2)*(2
*(20*a^3*d^3*f + 3*a^2*b*d^2*(45*d*e - 16*c*f) - 9*a*b^2*c*d*(7*d*e - 4*c*f) + 4
*b^3*c^2*(3*d*e - 2*c*f)) + 3*b*d*(21*a*b*d^2*e - 4*(b*c - a*d)*(3*b*d*e - 2*b*c
*f + 2*a*d*f))*x))/(315*d^4) - 2*a^3*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]]

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Rubi [A]  time = 0.781711, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 2 a^3 e \sqrt{c+d x}-2 a^3 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+\frac{2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )\right )}{315 d^4}+\frac{2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{21 d^2}+\frac{2 f (a+b x)^3 (c+d x)^{3/2}}{9 d} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^3*Sqrt[c + d*x]*(e + f*x))/x,x]

[Out]

2*a^3*e*Sqrt[c + d*x] + (2*(3*b*d*e - 2*b*c*f + 2*a*d*f)*(a + b*x)^2*(c + d*x)^(
3/2))/(21*d^2) + (2*f*(a + b*x)^3*(c + d*x)^(3/2))/(9*d) + (2*(c + d*x)^(3/2)*(2
*(20*a^3*d^3*f + 3*a^2*b*d^2*(45*d*e - 16*c*f) - 9*a*b^2*c*d*(7*d*e - 4*c*f) + 4
*b^3*c^2*(3*d*e - 2*c*f)) + 3*b*d*(21*a*b*d^2*e - 4*(b*c - a*d)*(3*b*d*e - 2*b*c
*f + 2*a*d*f))*x))/(315*d^4) - 2*a^3*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]]

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Rubi in Sympy [A]  time = 60.5883, size = 252, normalized size = 1.11 \[ - 2 a^{3} \sqrt{c} e \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )} + 2 a^{3} e \sqrt{c + d x} + \frac{2 f \left (a + b x\right )^{3} \left (c + d x\right )^{\frac{3}{2}}}{9 d} + \frac{4 \left (a + b x\right )^{2} \left (c + d x\right )^{\frac{3}{2}} \left (\frac{3 b d e}{2} + f \left (a d - b c\right )\right )}{21 d^{2}} + \frac{16 \left (c + d x\right )^{\frac{3}{2}} \left (15 a^{3} d^{3} f - 36 a^{2} b c d^{2} f + \frac{405 a^{2} b d^{3} e}{4} + 27 a b^{2} c^{2} d f - \frac{189 a b^{2} c d^{2} e}{4} - 6 b^{3} c^{3} f + 9 b^{3} c^{2} d e + \frac{9 b d x \left (21 a b d^{2} e + \left (4 a d - 4 b c\right ) \left (3 b d e + 2 f \left (a d - b c\right )\right )\right )}{8}\right )}{945 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(f*x+e)*(d*x+c)**(1/2)/x,x)

[Out]

-2*a**3*sqrt(c)*e*atanh(sqrt(c + d*x)/sqrt(c)) + 2*a**3*e*sqrt(c + d*x) + 2*f*(a
+ b*x)**3*(c + d*x)**(3/2)/(9*d) + 4*(a + b*x)**2*(c + d*x)**(3/2)*(3*b*d*e/2 +
f*(a*d - b*c))/(21*d**2) + 16*(c + d*x)**(3/2)*(15*a**3*d**3*f - 36*a**2*b*c*d*
*2*f + 405*a**2*b*d**3*e/4 + 27*a*b**2*c**2*d*f - 189*a*b**2*c*d**2*e/4 - 6*b**3
*c**3*f + 9*b**3*c**2*d*e + 9*b*d*x*(21*a*b*d**2*e + (4*a*d - 4*b*c)*(3*b*d*e +
2*f*(a*d - b*c)))/8)/(945*d**4)

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Mathematica [A]  time = 0.606165, size = 197, normalized size = 0.87 \[ \frac{2 \sqrt{c+d x} \left (105 a^3 d^3 (c f+3 d e+d f x)+63 a^2 b d^2 (c+d x) (-2 c f+5 d e+3 d f x)+9 a b^2 d (c+d x) \left (8 c^2 f-2 c d (7 e+6 f x)+3 d^2 x (7 e+5 f x)\right )+b^3 (-(c+d x)) \left (16 c^3 f-24 c^2 d (e+f x)+6 c d^2 x (6 e+5 f x)-5 d^3 x^2 (9 e+7 f x)\right )\right )}{315 d^4}-2 a^3 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^3*Sqrt[c + d*x]*(e + f*x))/x,x]

[Out]

(2*Sqrt[c + d*x]*(105*a^3*d^3*(3*d*e + c*f + d*f*x) + 63*a^2*b*d^2*(c + d*x)*(5*
d*e - 2*c*f + 3*d*f*x) + 9*a*b^2*d*(c + d*x)*(8*c^2*f + 3*d^2*x*(7*e + 5*f*x) -
2*c*d*(7*e + 6*f*x)) - b^3*(c + d*x)*(16*c^3*f - 24*c^2*d*(e + f*x) + 6*c*d^2*x*
(6*e + 5*f*x) - 5*d^3*x^2*(9*e + 7*f*x))))/(315*d^4) - 2*a^3*Sqrt[c]*e*ArcTanh[S
qrt[c + d*x]/Sqrt[c]]

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Maple [A]  time = 0.033, size = 301, normalized size = 1.3 \[ 2\,{\frac{1}{{d}^{4}} \left ( 1/9\,f{b}^{3} \left ( dx+c \right ) ^{9/2}+3/7\, \left ( dx+c \right ) ^{7/2}a{b}^{2}df-3/7\, \left ( dx+c \right ) ^{7/2}{b}^{3}cf+1/7\, \left ( dx+c \right ) ^{7/2}{b}^{3}de+3/5\, \left ( dx+c \right ) ^{5/2}{a}^{2}b{d}^{2}f-6/5\, \left ( dx+c \right ) ^{5/2}a{b}^{2}cdf+3/5\, \left ( dx+c \right ) ^{5/2}a{b}^{2}{d}^{2}e+3/5\, \left ( dx+c \right ) ^{5/2}{b}^{3}{c}^{2}f-2/5\, \left ( dx+c \right ) ^{5/2}{b}^{3}cde+1/3\, \left ( dx+c \right ) ^{3/2}{a}^{3}{d}^{3}f- \left ( dx+c \right ) ^{3/2}{a}^{2}bc{d}^{2}f+ \left ( dx+c \right ) ^{3/2}{a}^{2}b{d}^{3}e+ \left ( dx+c \right ) ^{3/2}a{b}^{2}{c}^{2}df- \left ( dx+c \right ) ^{3/2}a{b}^{2}c{d}^{2}e-1/3\, \left ( dx+c \right ) ^{3/2}{b}^{3}{c}^{3}f+1/3\, \left ( dx+c \right ) ^{3/2}{b}^{3}{c}^{2}de+{a}^{3}{d}^{4}e\sqrt{dx+c}-{a}^{3}\sqrt{c}{d}^{4}e{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(f*x+e)*(d*x+c)^(1/2)/x,x)

[Out]

2/d^4*(1/9*f*b^3*(d*x+c)^(9/2)+3/7*(d*x+c)^(7/2)*a*b^2*d*f-3/7*(d*x+c)^(7/2)*b^3
*c*f+1/7*(d*x+c)^(7/2)*b^3*d*e+3/5*(d*x+c)^(5/2)*a^2*b*d^2*f-6/5*(d*x+c)^(5/2)*a
*b^2*c*d*f+3/5*(d*x+c)^(5/2)*a*b^2*d^2*e+3/5*(d*x+c)^(5/2)*b^3*c^2*f-2/5*(d*x+c)
^(5/2)*b^3*c*d*e+1/3*(d*x+c)^(3/2)*a^3*d^3*f-(d*x+c)^(3/2)*a^2*b*c*d^2*f+(d*x+c)
^(3/2)*a^2*b*d^3*e+(d*x+c)^(3/2)*a*b^2*c^2*d*f-(d*x+c)^(3/2)*a*b^2*c*d^2*e-1/3*(
d*x+c)^(3/2)*b^3*c^3*f+1/3*(d*x+c)^(3/2)*b^3*c^2*d*e+a^3*d^4*e*(d*x+c)^(1/2)-a^3
*c^(1/2)*d^4*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)^3*sqrt(d*x + c)*(f*x + e)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245202, size = 1, normalized size = 0. \[ \left [\frac{315 \, a^{3} \sqrt{c} d^{4} e \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \,{\left (35 \, b^{3} d^{4} f x^{4} + 5 \,{\left (9 \, b^{3} d^{4} e +{\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \,{\left (3 \,{\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e -{\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \,{\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e -{\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f -{\left (3 \,{\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e -{\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt{d x + c}}{315 \, d^{4}}, -\frac{2 \,{\left (315 \, a^{3} \sqrt{-c} d^{4} e \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) -{\left (35 \, b^{3} d^{4} f x^{4} + 5 \,{\left (9 \, b^{3} d^{4} e +{\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \,{\left (3 \,{\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e -{\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \,{\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e -{\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f -{\left (3 \,{\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e -{\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt{d x + c}\right )}}{315 \, d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*sqrt(d*x + c)*(f*x + e)/x,x, algorithm="fricas")

[Out]

[1/315*(315*a^3*sqrt(c)*d^4*e*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(
35*b^3*d^4*f*x^4 + 5*(9*b^3*d^4*e + (b^3*c*d^3 + 27*a*b^2*d^4)*f)*x^3 + 3*(3*(b^
3*c*d^3 + 21*a*b^2*d^4)*e - (2*b^3*c^2*d^2 - 9*a*b^2*c*d^3 - 63*a^2*b*d^4)*f)*x^
2 + 3*(8*b^3*c^3*d - 42*a*b^2*c^2*d^2 + 105*a^2*b*c*d^3 + 105*a^3*d^4)*e - (16*b
^3*c^4 - 72*a*b^2*c^3*d + 126*a^2*b*c^2*d^2 - 105*a^3*c*d^3)*f - (3*(4*b^3*c^2*d
^2 - 21*a*b^2*c*d^3 - 105*a^2*b*d^4)*e - (8*b^3*c^3*d - 36*a*b^2*c^2*d^2 + 63*a^
2*b*c*d^3 + 105*a^3*d^4)*f)*x)*sqrt(d*x + c))/d^4, -2/315*(315*a^3*sqrt(-c)*d^4*
e*arctan(sqrt(d*x + c)/sqrt(-c)) - (35*b^3*d^4*f*x^4 + 5*(9*b^3*d^4*e + (b^3*c*d
^3 + 27*a*b^2*d^4)*f)*x^3 + 3*(3*(b^3*c*d^3 + 21*a*b^2*d^4)*e - (2*b^3*c^2*d^2 -
9*a*b^2*c*d^3 - 63*a^2*b*d^4)*f)*x^2 + 3*(8*b^3*c^3*d - 42*a*b^2*c^2*d^2 + 105*
a^2*b*c*d^3 + 105*a^3*d^4)*e - (16*b^3*c^4 - 72*a*b^2*c^3*d + 126*a^2*b*c^2*d^2
- 105*a^3*c*d^3)*f - (3*(4*b^3*c^2*d^2 - 21*a*b^2*c*d^3 - 105*a^2*b*d^4)*e - (8*
b^3*c^3*d - 36*a*b^2*c^2*d^2 + 63*a^2*b*c*d^3 + 105*a^3*d^4)*f)*x)*sqrt(d*x + c)
)/d^4]

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Sympy [A]  time = 117.726, size = 330, normalized size = 1.45 \[ - 2 a^{3} c e \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + d x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + d x \wedge - c < 0 \end{cases}\right ) + 2 a^{3} e \sqrt{c + d x} + \frac{2 b^{3} f \left (c + d x\right )^{\frac{9}{2}}}{9 d^{4}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}} \left (3 a b^{2} d f - 3 b^{3} c f + b^{3} d e\right )}{7 d^{4}} + \frac{2 \left (c + d x\right )^{\frac{5}{2}} \left (3 a^{2} b d^{2} f - 6 a b^{2} c d f + 3 a b^{2} d^{2} e + 3 b^{3} c^{2} f - 2 b^{3} c d e\right )}{5 d^{4}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a^{2} b d^{3} e + 3 a b^{2} c^{2} d f - 3 a b^{2} c d^{2} e - b^{3} c^{3} f + b^{3} c^{2} d e\right )}{3 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(f*x+e)*(d*x+c)**(1/2)/x,x)

[Out]

-2*a**3*c*e*Piecewise((-atan(sqrt(c + d*x)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(s
qrt(c + d*x)/sqrt(c))/sqrt(c), (-c < 0) & (c < c + d*x)), (atanh(sqrt(c + d*x)/s
qrt(c))/sqrt(c), (-c < 0) & (c > c + d*x))) + 2*a**3*e*sqrt(c + d*x) + 2*b**3*f*
(c + d*x)**(9/2)/(9*d**4) + 2*(c + d*x)**(7/2)*(3*a*b**2*d*f - 3*b**3*c*f + b**3
*d*e)/(7*d**4) + 2*(c + d*x)**(5/2)*(3*a**2*b*d**2*f - 6*a*b**2*c*d*f + 3*a*b**2
*d**2*e + 3*b**3*c**2*f - 2*b**3*c*d*e)/(5*d**4) + 2*(c + d*x)**(3/2)*(a**3*d**3
*f - 3*a**2*b*c*d**2*f + 3*a**2*b*d**3*e + 3*a*b**2*c**2*d*f - 3*a*b**2*c*d**2*e
- b**3*c**3*f + b**3*c**2*d*e)/(3*d**4)

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GIAC/XCAS [A]  time = 0.22605, size = 456, normalized size = 2.01 \[ \frac{2 \, a^{3} c \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) e}{\sqrt{-c}} + \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{3} d^{32} f - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{3} c d^{32} f + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{3} c^{2} d^{32} f - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c^{3} d^{32} f + 135 \,{\left (d x + c\right )}^{\frac{7}{2}} a b^{2} d^{33} f - 378 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{2} c d^{33} f + 315 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} c^{2} d^{33} f + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b d^{34} f - 315 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b c d^{34} f + 105 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} d^{35} f + 45 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{3} d^{33} e - 126 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{3} c d^{33} e + 105 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c^{2} d^{33} e + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{2} d^{34} e - 315 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} c d^{34} e + 315 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b d^{35} e + 315 \, \sqrt{d x + c} a^{3} d^{36} e\right )}}{315 \, d^{36}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*sqrt(d*x + c)*(f*x + e)/x,x, algorithm="giac")

[Out]

2*a^3*c*arctan(sqrt(d*x + c)/sqrt(-c))*e/sqrt(-c) + 2/315*(35*(d*x + c)^(9/2)*b^
3*d^32*f - 135*(d*x + c)^(7/2)*b^3*c*d^32*f + 189*(d*x + c)^(5/2)*b^3*c^2*d^32*f
- 105*(d*x + c)^(3/2)*b^3*c^3*d^32*f + 135*(d*x + c)^(7/2)*a*b^2*d^33*f - 378*(
d*x + c)^(5/2)*a*b^2*c*d^33*f + 315*(d*x + c)^(3/2)*a*b^2*c^2*d^33*f + 189*(d*x
+ c)^(5/2)*a^2*b*d^34*f - 315*(d*x + c)^(3/2)*a^2*b*c*d^34*f + 105*(d*x + c)^(3/
2)*a^3*d^35*f + 45*(d*x + c)^(7/2)*b^3*d^33*e - 126*(d*x + c)^(5/2)*b^3*c*d^33*e
+ 105*(d*x + c)^(3/2)*b^3*c^2*d^33*e + 189*(d*x + c)^(5/2)*a*b^2*d^34*e - 315*(
d*x + c)^(3/2)*a*b^2*c*d^34*e + 315*(d*x + c)^(3/2)*a^2*b*d^35*e + 315*sqrt(d*x
+ c)*a^3*d^36*e)/d^36