3.546 \(\int x \sqrt{a+b x} (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=221 \[ \frac{(5 a d+3 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{7/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+3 b c) (b c-a d)^2}{64 b^3 d^2}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (5 a d+3 b c) (b c-a d)}{32 b^3 d}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 b d} \]

[Out]

-((b*c - a*d)^2*(3*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^3*d^2) - ((b*
c - a*d)*(3*b*c + 5*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(32*b^3*d) - ((3*b*c + 5
*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(24*b^2*d) + ((a + b*x)^(3/2)*(c + d*x)^(
5/2))/(4*b*d) + ((b*c - a*d)^3*(3*b*c + 5*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(
Sqrt[b]*Sqrt[c + d*x])])/(64*b^(7/2)*d^(5/2))

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Rubi [A]  time = 0.322321, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(5 a d+3 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{7/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+3 b c) (b c-a d)^2}{64 b^3 d^2}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (5 a d+3 b c) (b c-a d)}{32 b^3 d}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 b d} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[a + b*x]*(c + d*x)^(3/2),x]

[Out]

-((b*c - a*d)^2*(3*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^3*d^2) - ((b*
c - a*d)*(3*b*c + 5*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(32*b^3*d) - ((3*b*c + 5
*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(24*b^2*d) + ((a + b*x)^(3/2)*(c + d*x)^(
5/2))/(4*b*d) + ((b*c - a*d)^3*(3*b*c + 5*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(
Sqrt[b]*Sqrt[c + d*x])])/(64*b^(7/2)*d^(5/2))

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Rubi in Sympy [A]  time = 34.853, size = 199, normalized size = 0.9 \[ \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}{4 b d} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (5 a d + 3 b c\right )}{24 b^{2} d} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (5 a d + 3 b c\right )}{32 b^{3} d} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (5 a d + 3 b c\right )}{64 b^{3} d^{2}} - \frac{\left (a d - b c\right )^{3} \left (5 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{64 b^{\frac{7}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(3/2)*(c + d*x)**(5/2)/(4*b*d) - (a + b*x)**(3/2)*(c + d*x)**(3/2)*(5
*a*d + 3*b*c)/(24*b**2*d) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)*(5*a*d +
3*b*c)/(32*b**3*d) - sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**2*(5*a*d + 3*b*c)/
(64*b**3*d**2) - (a*d - b*c)**3*(5*a*d + 3*b*c)*atanh(sqrt(b)*sqrt(c + d*x)/(sqr
t(d)*sqrt(a + b*x)))/(64*b**(7/2)*d**(5/2))

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Mathematica [A]  time = 0.192289, size = 194, normalized size = 0.88 \[ \frac{3 (b c-a d)^3 (5 a d+3 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )-2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x} \left (-15 a^3 d^3+a^2 b d^2 (31 c+10 d x)-a b^2 d \left (9 c^2+20 c d x+8 d^2 x^2\right )+b^3 \left (9 c^3-6 c^2 d x-72 c d^2 x^2-48 d^3 x^3\right )\right )}{384 b^{7/2} d^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[a + b*x]*(c + d*x)^(3/2),x]

[Out]

(-2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-15*a^3*d^3 + a^2*b*d^2*(31*c +
10*d*x) - a*b^2*d*(9*c^2 + 20*c*d*x + 8*d^2*x^2) + b^3*(9*c^3 - 6*c^2*d*x - 72*
c*d^2*x^2 - 48*d^3*x^3)) + 3*(b*c - a*d)^3*(3*b*c + 5*a*d)*Log[b*c + a*d + 2*b*d
*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(384*b^(7/2)*d^(5/2))

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Maple [B]  time = 0.02, size = 686, normalized size = 3.1 \[ -{\frac{1}{384\,{b}^{3}{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( -96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-16\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-144\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{4}-36\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}bc{d}^{3}+18\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{2}{c}^{2}{d}^{2}+12\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{3}{c}^{3}d-9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{4}{c}^{4}+20\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}b{d}^{3}-40\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{2}c{d}^{2}-12\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{3}{c}^{2}d-30\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}{d}^{3}+62\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}bc{d}^{2}-18\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}a{b}^{2}{c}^{2}d+18\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{3}{c}^{3} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-96*x^3*b^3*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1
/2)*(b*d)^(1/2)-16*x^2*a*b^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-144
*x^2*b^3*c*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+15*ln(1/2*(2*b*d*x+2*
(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*d^4-36*ln(
1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))
*a^3*b*c*d^3+18*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*
d+b*c)/(b*d)^(1/2))*a^2*b^2*c^2*d^2+12*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*
c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^3*c^3*d-9*ln(1/2*(2*b*d*x+2*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^4*c^4+20*(b*d)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b*d^3-40*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*
c*x+a*c)^(1/2)*x*a*b^2*c*d^2-12*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^
3*c^2*d-30*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*d^3+62*(b*d)^(1/2)*(b
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b*c*d^2-18*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*
c)^(1/2)*a*b^2*c^2*d+18*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^3*c^3)/(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)/b^3/d^2/(b*d)^(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)^(3/2)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.269836, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{3} x^{3} - 9 \, b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 31 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3} + 8 \,{\left (9 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (3 \, b^{3} c^{2} d + 10 \, a b^{2} c d^{2} - 5 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (3 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{768 \, \sqrt{b d} b^{3} d^{2}}, \frac{2 \,{\left (48 \, b^{3} d^{3} x^{3} - 9 \, b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 31 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3} + 8 \,{\left (9 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (3 \, b^{3} c^{2} d + 10 \, a b^{2} c d^{2} - 5 \, a^{2} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (3 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{384 \, \sqrt{-b d} b^{3} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*(d*x + c)^(3/2)*x,x, algorithm="fricas")

[Out]

[1/768*(4*(48*b^3*d^3*x^3 - 9*b^3*c^3 + 9*a*b^2*c^2*d - 31*a^2*b*c*d^2 + 15*a^3*
d^3 + 8*(9*b^3*c*d^2 + a*b^2*d^3)*x^2 + 2*(3*b^3*c^2*d + 10*a*b^2*c*d^2 - 5*a^2*
b*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(3*b^4*c^4 - 4*a*b^3*c^3*d -
6*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 - 5*a^4*d^4)*log(-4*(2*b^2*d^2*x + b^2*c*d +
a*b*d^2)*sqrt(b*x + a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a
^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x)*sqrt(b*d)))/(sqrt(b*d)*b^3*d^2), 1/384*(2*(48*
b^3*d^3*x^3 - 9*b^3*c^3 + 9*a*b^2*c^2*d - 31*a^2*b*c*d^2 + 15*a^3*d^3 + 8*(9*b^3
*c*d^2 + a*b^2*d^3)*x^2 + 2*(3*b^3*c^2*d + 10*a*b^2*c*d^2 - 5*a^2*b*d^3)*x)*sqrt
(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(3*b^4*c^4 - 4*a*b^3*c^3*d - 6*a^2*b^2*c^
2*d^2 + 12*a^3*b*c*d^3 - 5*a^4*d^4)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/
(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)*b^3*d^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.263997, size = 655, normalized size = 2.96 \[ \frac{\frac{10 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{2}} + \frac{b^{7} c d^{5} - 17 \, a b^{6} d^{6}}{b^{8} d^{6}}\right )} - \frac{5 \, b^{8} c^{2} d^{4} + 6 \, a b^{7} c d^{5} - 59 \, a^{2} b^{6} d^{6}}{b^{8} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{9} c^{3} d^{3} + a b^{8} c^{2} d^{4} - a^{2} b^{7} c d^{5} - 5 \, a^{3} b^{6} d^{6}\right )}}{b^{8} d^{6}}\right )} \sqrt{b x + a} + \frac{3 \,{\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b d^{3}}\right )} d{\left | b \right |}}{b^{2}} + \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{6} d^{2}} + \frac{b c d^{3} - 7 \, a d^{4}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{5} d^{4}}\right )} c{\left | b \right |}}{b^{3}}}{1920 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*(d*x + c)^(3/2)*x,x, algorithm="giac")

[Out]

1/1920*(10*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*
x + a)/b^2 + (b^7*c*d^5 - 17*a*b^6*d^6)/(b^8*d^6)) - (5*b^8*c^2*d^4 + 6*a*b^7*c*
d^5 - 59*a^2*b^6*d^6)/(b^8*d^6)) + 3*(5*b^9*c^3*d^3 + a*b^8*c^2*d^4 - a^2*b^7*c*
d^5 - 5*a^3*b^6*d^6)/(b^8*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2
*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4*d^4)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) +
sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^3))*d*abs(b)/b^2 + (sqrt(b^
2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/(b^6*d^2) +
(b*c*d^3 - 7*a*d^4)/(b^6*d^6)) - 3*(b^2*c^2*d^2 - a^2*d^4)/(b^6*d^6)) - 3*(b^3*
c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqr
t(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^5*d^4))*c*abs(b)/b^3)/b