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

Optimal. Leaf size=221 \[ -\frac{(3 a d+5 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+5 b c) (b c-a d)^2}{64 b^2 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (3 a d+5 b c) (b c-a d)}{96 b^2 d^2}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (3 a d+5 b c)}{24 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b d} \]

[Out]

((b*c - a*d)^2*(5*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^2*d^3) - ((b*c
- a*d)*(5*b*c + 3*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(96*b^2*d^2) - ((5*b*c +
3*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*b^2*d) + ((a + b*x)^(5/2)*(c + d*x)^(3
/2))/(4*b*d) - ((b*c - a*d)^3*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(S
qrt[b]*Sqrt[c + d*x])])/(64*b^(5/2)*d^(7/2))

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Rubi [A]  time = 0.327952, 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 \[ -\frac{(3 a d+5 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+5 b c) (b c-a d)^2}{64 b^2 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (3 a d+5 b c) (b c-a d)}{96 b^2 d^2}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (3 a d+5 b c)}{24 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b d} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*(5*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^2*d^3) - ((b*c
- a*d)*(5*b*c + 3*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(96*b^2*d^2) - ((5*b*c +
3*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*b^2*d) + ((a + b*x)^(5/2)*(c + d*x)^(3
/2))/(4*b*d) - ((b*c - a*d)^3*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(S
qrt[b]*Sqrt[c + d*x])])/(64*b^(5/2)*d^(7/2))

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Rubi in Sympy [A]  time = 34.7387, size = 201, normalized size = 0.91 \[ \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}}}{4 b d} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (3 a d + 5 b c\right )}{24 b^{2} d} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (3 a d + 5 b c\right )}{96 b^{2} d^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (3 a d + 5 b c\right )}{64 b^{2} d^{3}} + \frac{\left (a d - b c\right )^{3} \left (3 a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{5}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(5/2)*(c + d*x)**(3/2)/(4*b*d) - (a + b*x)**(5/2)*sqrt(c + d*x)*(3*a*
d + 5*b*c)/(24*b**2*d) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)*(3*a*d + 5*b
*c)/(96*b**2*d**2) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**2*(3*a*d + 5*b*c)/
(64*b**2*d**3) + (a*d - b*c)**3*(3*a*d + 5*b*c)*atanh(sqrt(d)*sqrt(a + b*x)/(sqr
t(b)*sqrt(c + d*x)))/(64*b**(5/2)*d**(7/2))

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Mathematica [A]  time = 0.190529, size = 194, normalized size = 0.88 \[ \frac{2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x} \left (-9 a^3 d^3+3 a^2 b d^2 (3 c+2 d x)+a b^2 d \left (-31 c^2+20 c d x+72 d^2 x^2\right )+b^3 \left (15 c^3-10 c^2 d x+8 c d^2 x^2+48 d^3 x^3\right )\right )-3 (b c-a d)^3 (3 a d+5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{384 b^{5/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^(3/2)*(d*x+c)^(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)+144*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)+16*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)+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))*a^4*d^4-12*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^
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+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*b^3*c^3*d-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))*b^4*c^4+12*(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^2-20*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^3*
c^2*d-18*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*d^3+18*(b*d)^(1/2)*(b*d
*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b*c*d^2-62*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)
^(1/2)*a*b^2*c^2*d+30*(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)/d^3/b^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((b*x + a)^(3/2)*sqrt(d*x + c)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24702, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 31 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 9 \, a^{3} d^{3} + 8 \,{\left (b^{3} c d^{2} + 9 \, a b^{2} d^{3}\right )} x^{2} - 2 \,{\left (5 \, b^{3} c^{2} d - 10 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (5 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{768 \, \sqrt{b d} b^{2} d^{3}}, \frac{2 \,{\left (48 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 31 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 9 \, a^{3} d^{3} + 8 \,{\left (b^{3} c d^{2} + 9 \, a b^{2} d^{3}\right )} x^{2} - 2 \,{\left (5 \, b^{3} c^{2} d - 10 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (5 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{384 \, \sqrt{-b d} b^{2} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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*(b*x+a)**(3/2)*(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.256552, size = 653, normalized size = 2.95 \[ \frac{\frac{10 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{2}} + \frac{b^{7} c d^{5} - 17 \, a b^{6} d^{6}}{b^{8} d^{6}}\right )} - \frac{5 \, b^{8} c^{2} d^{4} + 6 \, a b^{7} c d^{5} - 59 \, a^{2} b^{6} d^{6}}{b^{8} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{9} c^{3} d^{3} + a b^{8} c^{2} d^{4} - a^{2} b^{7} c d^{5} - 5 \, a^{3} b^{6} d^{6}\right )}}{b^{8} d^{6}}\right )} \sqrt{b x + a} + \frac{3 \,{\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b d^{3}}\right )}{\left | b \right |}}{b} + \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{6} d^{2}} + \frac{b c d^{3} - 7 \, a d^{4}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{5} d^{4}}\right )} a{\left | b \right |}}{b^{3}}}{1920 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)*sqrt(d*x + c)*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))*abs(b)/b + (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) + sqrt(b^
2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^5*d^4))*a*abs(b)/b^3)/b