3.1472 \(\int (a+b x)^{3/2} (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=189 \[ \frac{3 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{5/2}}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 b^2 d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}{32 b^2 d}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)}{8 b^2}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b} \]

[Out]

(-3*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^2*d^2) + ((b*c - a*d)^2*(a
+ b*x)^(3/2)*Sqrt[c + d*x])/(32*b^2*d) + ((b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d
*x])/(8*b^2) + ((a + b*x)^(5/2)*(c + d*x)^(3/2))/(4*b) + (3*(b*c - a*d)^4*ArcTan
h[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(5/2)*d^(5/2))

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Rubi [A]  time = 0.228303, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{3 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{5/2}}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 b^2 d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}{32 b^2 d}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)}{8 b^2}+\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^2*d^2) + ((b*c - a*d)^2*(a
+ b*x)^(3/2)*Sqrt[c + d*x])/(32*b^2*d) + ((b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d
*x])/(8*b^2) + ((a + b*x)^(5/2)*(c + d*x)^(3/2))/(4*b) + (3*(b*c - a*d)^4*ArcTan
h[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(5/2)*d^(5/2))

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Rubi in Sympy [A]  time = 34.8366, size = 167, normalized size = 0.88 \[ \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}{4 d} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (a d - b c\right )}{8 d^{2}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}}{32 b d^{2}} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3}}{64 b^{2} d^{2}} + \frac{3 \left (a d - b c\right )^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{64 b^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(3/2)*(c + d*x)**(5/2)/(4*d) + sqrt(a + b*x)*(c + d*x)**(5/2)*(a*d -
b*c)/(8*d**2) + sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)**2/(32*b*d**2) - 3*sq
rt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**3/(64*b**2*d**2) + 3*(a*d - b*c)**4*atanh
(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(a + b*x)))/(64*b**(5/2)*d**(5/2))

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Mathematica [A]  time = 0.194199, size = 180, normalized size = 0.95 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-3 a^3 d^3+a^2 b d^2 (11 c+2 d x)+a b^2 d \left (11 c^2+44 c d x+24 d^2 x^2\right )+b^3 \left (-3 c^3+2 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )\right )}{64 b^2 d^2}+\frac{3 (b c-a d)^4 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{5/2} d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-3*a^3*d^3 + a^2*b*d^2*(11*c + 2*d*x) + a*b^2*d*(1
1*c^2 + 44*c*d*x + 24*d^2*x^2) + b^3*(-3*c^3 + 2*c^2*d*x + 24*c*d^2*x^2 + 16*d^3
*x^3)))/(64*b^2*d^2) + (3*(b*c - a*d)^4*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt
[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(128*b^(5/2)*d^(5/2))

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Maple [B]  time = 0.01, size = 640, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/d*(b*x+a)^(3/2)*(d*x+c)^(5/2)+1/8/d*(b*x+a)^(1/2)*(d*x+c)^(5/2)*a-1/8/d^2*(b
*x+a)^(1/2)*(d*x+c)^(5/2)*b*c+1/32/b*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a^2-1/16/d*(d*x
+c)^(3/2)*(b*x+a)^(1/2)*a*c+1/32/d^2*(d*x+c)^(3/2)*(b*x+a)^(1/2)*c^2*b-3/64*d/b^
2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^3+9/64/b*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^2*c-9/64/
d*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a*c^2+3/64/d^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*c^3*b+3
/128*d^2/b^2*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2
*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^4-3/32*d/
b*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)
/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^3*c+9/64*((b*x+a)*(d
*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+
(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^2*c^2-3/32/d*((b*x+a)*(d*x+c))^(1
/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+
(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a*c^3*b+3/128/d^2*((b*x+a)*(d*x+c))^(1/2)/(d
*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b
*c)*x+a*c)^(1/2))/(b*d)^(1/2)*c^4*b^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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241143, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (16 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 11 \, a b^{2} c^{2} d + 11 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3} + 24 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (b^{3} c^{2} d + 22 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{256 \, \sqrt{b d} b^{2} d^{2}}, \frac{2 \,{\left (16 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 11 \, a b^{2} c^{2} d + 11 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3} + 24 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (b^{3} c^{2} d + 22 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{128 \, \sqrt{-b d} b^{2} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/256*(4*(16*b^3*d^3*x^3 - 3*b^3*c^3 + 11*a*b^2*c^2*d + 11*a^2*b*c*d^2 - 3*a^3*
d^3 + 24*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 2*(b^3*c^2*d + 22*a*b^2*c*d^2 + a^2*b*d^3
)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*
b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*s
qrt(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^2), 1/128*(2*(16*b^3*d^3*x^3
- 3*b^3*c^3 + 11*a*b^2*c^2*d + 11*a^2*b*c*d^2 - 3*a^3*d^3 + 24*(b^3*c*d^2 + a*b
^2*d^3)*x^2 + 2*(b^3*c^2*d + 22*a*b^2*c*d^2 + a^2*b*d^3)*x)*sqrt(-b*d)*sqrt(b*x
+ a)*sqrt(d*x + c) + 3*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*
d^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^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.308207, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done