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

Optimal. Leaf size=226 \[ -\frac{2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (3 c f+d e)+3 a b^2 c d (16 c f+21 d e)-5 b^3 c^2 (4 c f+27 d e)\right )-3 b d x \left (4 (b c-a d) (-2 a d f+2 b c f+3 b d e)+21 b^2 c d e\right )\right )}{315 b^4}+\frac{2 (a+b x)^{3/2} (c+d x)^2 (-2 a d f+2 b c f+3 b d e)}{21 b^2}+2 c^3 e \sqrt{a+b x}-2 \sqrt{a} c^3 e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2 f (a+b x)^{3/2} (c+d x)^3}{9 b} \]

[Out]

2*c^3*e*Sqrt[a + b*x] + (2*(3*b*d*e + 2*b*c*f - 2*a*d*f)*(a + b*x)^(3/2)*(c + d*
x)^2)/(21*b^2) + (2*f*(a + b*x)^(3/2)*(c + d*x)^3)/(9*b) - (2*(a + b*x)^(3/2)*(2
*(8*a^3*d^3*f - 12*a^2*b*d^2*(d*e + 3*c*f) - 5*b^3*c^2*(27*d*e + 4*c*f) + 3*a*b^
2*c*d*(21*d*e + 16*c*f)) - 3*b*d*(21*b^2*c*d*e + 4*(b*c - a*d)*(3*b*d*e + 2*b*c*
f - 2*a*d*f))*x))/(315*b^4) - 2*Sqrt[a]*c^3*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Rubi [A]  time = 0.755811, antiderivative size = 226, 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 \[ -\frac{2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (3 c f+d e)+3 a b^2 c d (16 c f+21 d e)-5 b^3 c^2 (4 c f+27 d e)\right )-3 b d x \left (4 (b c-a d) (-2 a d f+2 b c f+3 b d e)+21 b^2 c d e\right )\right )}{315 b^4}+\frac{2 (a+b x)^{3/2} (c+d x)^2 (-2 a d f+2 b c f+3 b d e)}{21 b^2}+2 c^3 e \sqrt{a+b x}-2 \sqrt{a} c^3 e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2 f (a+b x)^{3/2} (c+d x)^3}{9 b} \]

Antiderivative was successfully verified.

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

[Out]

2*c^3*e*Sqrt[a + b*x] + (2*(3*b*d*e + 2*b*c*f - 2*a*d*f)*(a + b*x)^(3/2)*(c + d*
x)^2)/(21*b^2) + (2*f*(a + b*x)^(3/2)*(c + d*x)^3)/(9*b) - (2*(a + b*x)^(3/2)*(2
*(8*a^3*d^3*f - 12*a^2*b*d^2*(d*e + 3*c*f) - 5*b^3*c^2*(27*d*e + 4*c*f) + 3*a*b^
2*c*d*(21*d*e + 16*c*f)) - 3*b*d*(21*b^2*c*d*e + 4*(b*c - a*d)*(3*b*d*e + 2*b*c*
f - 2*a*d*f))*x))/(315*b^4) - 2*Sqrt[a]*c^3*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Rubi in Sympy [A]  time = 60.5421, size = 252, normalized size = 1.12 \[ - 2 \sqrt{a} c^{3} e \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 2 c^{3} e \sqrt{a + b x} + \frac{2 f \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{3}}{9 b} - \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{2} \left (- \frac{3 b d e}{2} + f \left (a d - b c\right )\right )}{21 b^{2}} - \frac{16 \left (a + b x\right )^{\frac{3}{2}} \left (6 a^{3} d^{3} f - 27 a^{2} b c d^{2} f - 9 a^{2} b d^{3} e + 36 a b^{2} c^{2} d f + \frac{189 a b^{2} c d^{2} e}{4} - 15 b^{3} c^{3} f - \frac{405 b^{3} c^{2} d e}{4} - \frac{9 b d x \left (21 b^{2} c d e + \left (4 a d - 4 b c\right ) \left (- 3 b d e + 2 f \left (a d - b c\right )\right )\right )}{8}\right )}{945 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.614682, size = 236, normalized size = 1.04 \[ \frac{2 \sqrt{a+b x} \left (-16 a^4 d^3 f+8 a^3 b d^2 (9 c f+3 d e+d f x)-6 a^2 b^2 d \left (21 c^2 f+3 c d (7 e+2 f x)+d^2 x (2 e+f x)\right )+a b^3 \left (105 c^3 f+63 c^2 d (5 e+f x)+9 c d^2 x (7 e+3 f x)+d^3 x^2 (9 e+5 f x)\right )+b^4 \left (105 c^3 (3 e+f x)+63 c^2 d x (5 e+3 f x)+27 c d^2 x^2 (7 e+5 f x)+5 d^3 x^3 (9 e+7 f x)\right )\right )}{315 b^4}-2 \sqrt{a} c^3 e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x]*(-16*a^4*d^3*f + 8*a^3*b*d^2*(3*d*e + 9*c*f + d*f*x) - 6*a^2*b^
2*d*(21*c^2*f + d^2*x*(2*e + f*x) + 3*c*d*(7*e + 2*f*x)) + a*b^3*(105*c^3*f + 63
*c^2*d*(5*e + f*x) + 9*c*d^2*x*(7*e + 3*f*x) + d^3*x^2*(9*e + 5*f*x)) + b^4*(105
*c^3*(3*e + f*x) + 63*c^2*d*x*(5*e + 3*f*x) + 27*c*d^2*x^2*(7*e + 5*f*x) + 5*d^3
*x^3*(9*e + 7*f*x))))/(315*b^4) - 2*Sqrt[a]*c^3*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Maple [A]  time = 0.016, size = 301, normalized size = 1.3 \[ 2\,{\frac{1}{{b}^{4}} \left ( 1/9\,f{d}^{3} \left ( bx+a \right ) ^{9/2}-3/7\, \left ( bx+a \right ) ^{7/2}a{d}^{3}f+3/7\, \left ( bx+a \right ) ^{7/2}bc{d}^{2}f+1/7\, \left ( bx+a \right ) ^{7/2}b{d}^{3}e+3/5\, \left ( bx+a \right ) ^{5/2}{a}^{2}{d}^{3}f-6/5\, \left ( bx+a \right ) ^{5/2}abc{d}^{2}f-2/5\, \left ( bx+a \right ) ^{5/2}ab{d}^{3}e+3/5\, \left ( bx+a \right ) ^{5/2}{b}^{2}{c}^{2}df+3/5\, \left ( bx+a \right ) ^{5/2}{b}^{2}c{d}^{2}e-1/3\, \left ( bx+a \right ) ^{3/2}{a}^{3}{d}^{3}f+ \left ( bx+a \right ) ^{3/2}{a}^{2}bc{d}^{2}f+1/3\, \left ( bx+a \right ) ^{3/2}{a}^{2}b{d}^{3}e- \left ( bx+a \right ) ^{3/2}a{b}^{2}{c}^{2}df- \left ( bx+a \right ) ^{3/2}a{b}^{2}c{d}^{2}e+1/3\, \left ( bx+a \right ) ^{3/2}{b}^{3}{c}^{3}f+ \left ( bx+a \right ) ^{3/2}{b}^{3}{c}^{2}de+{b}^{4}{c}^{3}e\sqrt{bx+a}-\sqrt{a}{b}^{4}{c}^{3}e{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230744, size = 1, normalized size = 0. \[ \left [\frac{315 \, \sqrt{a} b^{4} c^{3} e \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (35 \, b^{4} d^{3} f x^{4} + 5 \,{\left (9 \, b^{4} d^{3} e +{\left (27 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f\right )} x^{3} + 3 \,{\left (3 \,{\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e +{\left (63 \, b^{4} c^{2} d + 9 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} f\right )} x^{2} + 3 \,{\left (105 \, b^{4} c^{3} + 105 \, a b^{3} c^{2} d - 42 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e +{\left (105 \, a b^{3} c^{3} - 126 \, a^{2} b^{2} c^{2} d + 72 \, a^{3} b c d^{2} - 16 \, a^{4} d^{3}\right )} f +{\left (3 \,{\left (105 \, b^{4} c^{2} d + 21 \, a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3}\right )} e +{\left (105 \, b^{4} c^{3} + 63 \, a b^{3} c^{2} d - 36 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} f\right )} x\right )} \sqrt{b x + a}}{315 \, b^{4}}, -\frac{2 \,{\left (315 \, \sqrt{-a} b^{4} c^{3} e \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (35 \, b^{4} d^{3} f x^{4} + 5 \,{\left (9 \, b^{4} d^{3} e +{\left (27 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f\right )} x^{3} + 3 \,{\left (3 \,{\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e +{\left (63 \, b^{4} c^{2} d + 9 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} f\right )} x^{2} + 3 \,{\left (105 \, b^{4} c^{3} + 105 \, a b^{3} c^{2} d - 42 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e +{\left (105 \, a b^{3} c^{3} - 126 \, a^{2} b^{2} c^{2} d + 72 \, a^{3} b c d^{2} - 16 \, a^{4} d^{3}\right )} f +{\left (3 \,{\left (105 \, b^{4} c^{2} d + 21 \, a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3}\right )} e +{\left (105 \, b^{4} c^{3} + 63 \, a b^{3} c^{2} d - 36 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} f\right )} x\right )} \sqrt{b x + a}\right )}}{315 \, b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/315*(315*sqrt(a)*b^4*c^3*e*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(
35*b^4*d^3*f*x^4 + 5*(9*b^4*d^3*e + (27*b^4*c*d^2 + a*b^3*d^3)*f)*x^3 + 3*(3*(21
*b^4*c*d^2 + a*b^3*d^3)*e + (63*b^4*c^2*d + 9*a*b^3*c*d^2 - 2*a^2*b^2*d^3)*f)*x^
2 + 3*(105*b^4*c^3 + 105*a*b^3*c^2*d - 42*a^2*b^2*c*d^2 + 8*a^3*b*d^3)*e + (105*
a*b^3*c^3 - 126*a^2*b^2*c^2*d + 72*a^3*b*c*d^2 - 16*a^4*d^3)*f + (3*(105*b^4*c^2
*d + 21*a*b^3*c*d^2 - 4*a^2*b^2*d^3)*e + (105*b^4*c^3 + 63*a*b^3*c^2*d - 36*a^2*
b^2*c*d^2 + 8*a^3*b*d^3)*f)*x)*sqrt(b*x + a))/b^4, -2/315*(315*sqrt(-a)*b^4*c^3*
e*arctan(sqrt(b*x + a)/sqrt(-a)) - (35*b^4*d^3*f*x^4 + 5*(9*b^4*d^3*e + (27*b^4*
c*d^2 + a*b^3*d^3)*f)*x^3 + 3*(3*(21*b^4*c*d^2 + a*b^3*d^3)*e + (63*b^4*c^2*d +
9*a*b^3*c*d^2 - 2*a^2*b^2*d^3)*f)*x^2 + 3*(105*b^4*c^3 + 105*a*b^3*c^2*d - 42*a^
2*b^2*c*d^2 + 8*a^3*b*d^3)*e + (105*a*b^3*c^3 - 126*a^2*b^2*c^2*d + 72*a^3*b*c*d
^2 - 16*a^4*d^3)*f + (3*(105*b^4*c^2*d + 21*a*b^3*c*d^2 - 4*a^2*b^2*d^3)*e + (10
5*b^4*c^3 + 63*a*b^3*c^2*d - 36*a^2*b^2*c*d^2 + 8*a^3*b*d^3)*f)*x)*sqrt(b*x + a)
)/b^4]

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Sympy [A]  time = 94.6184, size = 330, normalized size = 1.46 \[ - 2 a c^{3} e \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) + 2 c^{3} e \sqrt{a + b x} + \frac{2 d^{3} f \left (a + b x\right )^{\frac{9}{2}}}{9 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (- 3 a d^{3} f + 3 b c d^{2} f + b d^{3} e\right )}{7 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (3 a^{2} d^{3} f - 6 a b c d^{2} f - 2 a b d^{3} e + 3 b^{2} c^{2} d f + 3 b^{2} c d^{2} e\right )}{5 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (- a^{3} d^{3} f + 3 a^{2} b c d^{2} f + a^{2} b d^{3} e - 3 a b^{2} c^{2} d f - 3 a b^{2} c d^{2} e + b^{3} c^{3} f + 3 b^{3} c^{2} d e\right )}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*c**3*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**3*e*sqrt(a + b*x) + 2*d**3*f*
(a + b*x)**(9/2)/(9*b**4) + 2*(a + b*x)**(7/2)*(-3*a*d**3*f + 3*b*c*d**2*f + b*d
**3*e)/(7*b**4) + 2*(a + b*x)**(5/2)*(3*a**2*d**3*f - 6*a*b*c*d**2*f - 2*a*b*d**
3*e + 3*b**2*c**2*d*f + 3*b**2*c*d**2*e)/(5*b**4) + 2*(a + b*x)**(3/2)*(-a**3*d*
*3*f + 3*a**2*b*c*d**2*f + 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 + 3*b**3*c**2*d*e)/(3*b**4)

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GIAC/XCAS [A]  time = 0.226678, size = 456, normalized size = 2.02 \[ \frac{2 \, a c^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) e}{\sqrt{-a}} + \frac{2 \,{\left (105 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{35} c^{3} f + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{34} c^{2} d f - 315 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{34} c^{2} d f + 135 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{33} c d^{2} f - 378 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{33} c d^{2} f + 315 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{33} c d^{2} f + 35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{32} d^{3} f - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{32} d^{3} f + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{32} d^{3} f - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{32} d^{3} f + 315 \, \sqrt{b x + a} b^{36} c^{3} e + 315 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{35} c^{2} d e + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{34} c d^{2} e - 315 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{34} c d^{2} e + 45 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{33} d^{3} e - 126 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{33} d^{3} e + 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{33} d^{3} e\right )}}{315 \, b^{36}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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