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

Optimal. Leaf size=258 \[ -\frac{3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{7/2} d^{7/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{128 b^3 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c) (b c-a d)^2}{64 b^3 d^2}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (a d+b c) (b c-a d)}{16 b^3 d}-\frac{(a+b x)^{5/2} (c+d x)^{3/2} (a d+b c)}{8 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d} \]

[Out]

(3*(b*c - a*d)^3*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^3*d^3) - ((b*c
- a*d)^2*(b*c + a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(64*b^3*d^2) - ((b*c - a*d)*
(b*c + a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(16*b^3*d) - ((b*c + a*d)*(a + b*x)^(
5/2)*(c + d*x)^(3/2))/(8*b^2*d) + ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(5*b*d) - (3
*(b*c - a*d)^4*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x
])])/(128*b^(7/2)*d^(7/2))

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Rubi [A]  time = 0.393207, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{7/2} d^{7/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{128 b^3 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c) (b c-a d)^2}{64 b^3 d^2}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (a d+b c) (b c-a d)}{16 b^3 d}-\frac{(a+b x)^{5/2} (c+d x)^{3/2} (a d+b c)}{8 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 b d} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b*c - a*d)^3*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^3*d^3) - ((b*c
- a*d)^2*(b*c + a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(64*b^3*d^2) - ((b*c - a*d)*
(b*c + a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(16*b^3*d) - ((b*c + a*d)*(a + b*x)^(
5/2)*(c + d*x)^(3/2))/(8*b^2*d) + ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(5*b*d) - (3
*(b*c - a*d)^4*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x
])])/(128*b^(7/2)*d^(7/2))

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Rubi in Sympy [A]  time = 47.4236, size = 230, normalized size = 0.89 \[ \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}}}{5 b d} - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d + b c\right )}{8 b^{2} d} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (a d + b c\right )}{16 b^{3} d} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (a d + b c\right )}{64 b^{3} d^{2}} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (a d + b c\right )}{128 b^{3} d^{3}} - \frac{3 \left (a d - b c\right )^{4} \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{128 b^{\frac{7}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(5/2)*(c + d*x)**(5/2)/(5*b*d) - (a + b*x)**(5/2)*(c + d*x)**(3/2)*(a
*d + b*c)/(8*b**2*d) + (a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*c)*(a*d + b*c)/(1
6*b**3*d) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**2*(a*d + b*c)/(64*b**3*d
**2) - 3*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**3*(a*d + b*c)/(128*b**3*d**3)
- 3*(a*d - b*c)**4*(a*d + b*c)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(a + b*x
)))/(128*b**(7/2)*d**(7/2))

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Mathematica [A]  time = 0.236855, size = 240, normalized size = 0.93 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^4 d^4-10 a^3 b d^3 (4 c+d x)+2 a^2 b^2 d^2 \left (9 c^2+13 c d x+4 d^2 x^2\right )+2 a b^3 d \left (-20 c^3+13 c^2 d x+136 c d^2 x^2+88 d^3 x^3\right )+b^4 \left (15 c^4-10 c^3 d x+8 c^2 d^2 x^2+176 c d^3 x^3+128 d^4 x^4\right )\right )}{640 b^3 d^3}-\frac{3 (b c-a d)^4 (a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{7/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*a^4*d^4 - 10*a^3*b*d^3*(4*c + d*x) + 2*a^2*b^2*
d^2*(9*c^2 + 13*c*d*x + 4*d^2*x^2) + 2*a*b^3*d*(-20*c^3 + 13*c^2*d*x + 136*c*d^2
*x^2 + 88*d^3*x^3) + b^4*(15*c^4 - 10*c^3*d*x + 8*c^2*d^2*x^2 + 176*c*d^3*x^3 +
128*d^4*x^4)))/(640*b^3*d^3) - (3*(b*c - a*d)^4*(b*c + a*d)*Log[b*c + a*d + 2*b*
d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(256*b^(7/2)*d^(7/2))

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Maple [B]  time = 0.02, size = 942, normalized size = 3.7 \[ -{\frac{1}{1280\,{b}^{3}{d}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( -256\,{x}^{4}{b}^{4}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-352\,{x}^{3}a{b}^{3}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-352\,{x}^{3}{b}^{4}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-16\,{x}^{2}{a}^{2}{b}^{2}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-544\,{x}^{2}a{b}^{3}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-16\,{x}^{2}{b}^{4}{c}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{5}{d}^{5}-45\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}bc{d}^{4}+30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{b}^{2}{c}^{2}{d}^{3}+30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{3}{c}^{3}{d}^{2}-45\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{4}{c}^{4}d+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{5}{c}^{5}+20\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{d}^{4}-52\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}{b}^{2}c{d}^{3}-52\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{3}{c}^{2}{d}^{2}+20\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{4}{c}^{3}d-30\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{d}^{4}+80\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}bc{d}^{3}-36\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+80\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}a{b}^{3}{c}^{3}d-30\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{4}{c}^{4} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1280*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-256*x^4*b^4*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2)*(b*d)^(1/2)-352*x^3*a*b^3*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-
352*x^3*b^4*c*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-16*x^2*a^2*b^2*d^4
*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-544*x^2*a*b^3*c*d^3*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)*(b*d)^(1/2)-16*x^2*b^4*c^2*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^5*d^5-45*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*b*c*d^4+30*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^2*c^2*d^3+30*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^3*c^3*d^2-45*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^4*c^4*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^5*c^5+20*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*
x+a*c)^(1/2)*x*a^3*b*d^4-52*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b^
2*c*d^3-52*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b^3*c^2*d^2+20*(b*d)^
(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^4*c^3*d-30*(b*d)^(1/2)*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)*a^4*d^4+80*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c*
d^3-36*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^2*c^2*d^2+80*(b*d)^(1/2
)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^3*d-30*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*
x+a*c)^(1/2)*b^4*c^4)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/b^3/d^3/(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)*(d*x + c)^(3/2)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275236, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (128 \, b^{4} d^{4} x^{4} + 15 \, b^{4} c^{4} - 40 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 40 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4} + 176 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (b^{4} c^{2} d^{2} + 34 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} - 2 \,{\left (5 \, b^{4} c^{3} d - 13 \, a b^{3} c^{2} d^{2} - 13 \, a^{2} b^{2} c d^{3} + 5 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{2560 \, \sqrt{b d} b^{3} d^{3}}, \frac{2 \,{\left (128 \, b^{4} d^{4} x^{4} + 15 \, b^{4} c^{4} - 40 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 40 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4} + 176 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (b^{4} c^{2} d^{2} + 34 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} - 2 \,{\left (5 \, b^{4} c^{3} d - 13 \, a b^{3} c^{2} d^{2} - 13 \, a^{2} b^{2} c d^{3} + 5 \, a^{3} b d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{1280 \, \sqrt{-b d} b^{3} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2560*(4*(128*b^4*d^4*x^4 + 15*b^4*c^4 - 40*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 -
40*a^3*b*c*d^3 + 15*a^4*d^4 + 176*(b^4*c*d^3 + a*b^3*d^4)*x^3 + 8*(b^4*c^2*d^2
+ 34*a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 - 2*(5*b^4*c^3*d - 13*a*b^3*c^2*d^2 - 13*a^2
*b^2*c*d^3 + 5*a^3*b*d^4)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(b^5*c^5
- 3*a*b^4*c^4*d + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - 3*a^4*b*c*d^4 + a^5*d
^5)*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^3), 1/1280*(2*(128*b^4*d^4*x^4 + 15*b^4*c^4 - 40*a*b^3*c^3*d +
18*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 + 15*a^4*d^4 + 176*(b^4*c*d^3 + a*b^3*d^4)*
x^3 + 8*(b^4*c^2*d^2 + 34*a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 - 2*(5*b^4*c^3*d - 13*a
*b^3*c^2*d^2 - 13*a^2*b^2*c*d^3 + 5*a^3*b*d^4)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(
d*x + c) - 15*(b^5*c^5 - 3*a*b^4*c^4*d + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 -
3*a^4*b*c*d^4 + a^5*d^5)*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^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.304079, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done