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

Optimal. Leaf size=154 \[ \frac{(b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{12 b d}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b} \]

[Out]

-((b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b*d^2) + ((b*c - a*d)*(a + b*x)^
(3/2)*Sqrt[c + d*x])/(12*b*d) + ((a + b*x)^(5/2)*Sqrt[c + d*x])/(3*b) + ((b*c -
a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*b^(3/2)*d^(5
/2))

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Rubi [A]  time = 0.181701, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{(b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{12 b d}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b*d^2) + ((b*c - a*d)*(a + b*x)^
(3/2)*Sqrt[c + d*x])/(12*b*d) + ((a + b*x)^(5/2)*Sqrt[c + d*x])/(3*b) + ((b*c -
a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*b^(3/2)*d^(5
/2))

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Rubi in Sympy [A]  time = 23.0523, size = 129, normalized size = 0.84 \[ \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{3 d} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{4 d^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2}}{8 b d^{2}} - \frac{\left (a d - b c\right )^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{8 b^{\frac{3}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.11933, size = 141, normalized size = 0.92 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2+2 a b d (4 c+7 d x)+b^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )\right )}{24 b d^2}+\frac{(b c-a d)^3 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{3/2} d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(3*a^2*d^2 + 2*a*b*d*(4*c + 7*d*x) + b^2*(-3*c^2 +
2*c*d*x + 8*d^2*x^2)))/(24*b*d^2) + ((b*c - a*d)^3*Log[b*c + a*d + 2*b*d*x + 2*S
qrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(16*b^(3/2)*d^(5/2))

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Maple [B]  time = 0.007, size = 460, normalized size = 3. \[{\frac{1}{3\,d} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{a}{4\,d}\sqrt{bx+a} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{bc}{4\,{d}^{2}}\sqrt{bx+a} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}}{8\,b}\sqrt{bx+a}\sqrt{dx+c}}-{\frac{ac}{4\,d}\sqrt{bx+a}\sqrt{dx+c}}+{\frac{{c}^{2}b}{8\,{d}^{2}}\sqrt{bx+a}\sqrt{dx+c}}-{\frac{{a}^{3}d}{16\,b}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}+{\frac{3\,{a}^{2}c}{16}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}-{\frac{3\,a{c}^{2}b}{16\,d}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}+{\frac{{c}^{3}{b}^{2}}{16\,{d}^{2}}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/d*(b*x+a)^(3/2)*(d*x+c)^(3/2)+1/4/d*(b*x+a)^(1/2)*(d*x+c)^(3/2)*a-1/4/d^2*(b
*x+a)^(1/2)*(d*x+c)^(3/2)*b*c+1/8/b*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^2-1/4/d*(d*x+c
)^(1/2)*(b*x+a)^(1/2)*a*c+1/8/d^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*c^2*b-1/16*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+3/16*((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-3/16/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^2*b+1/16/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^3*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)*sqrt(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233757, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 8 \, a b c d + 3 \, a^{2} d^{2} + 2 \,{\left (b^{2} c d + 7 \, a b d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{96 \, \sqrt{b d} b d^{2}}, \frac{2 \,{\left (8 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 8 \, a b c d + 3 \, a^{2} d^{2} + 2 \,{\left (b^{2} c d + 7 \, a b d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{48 \, \sqrt{-b d} b d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*b^2*d^2*x^2 - 3*b^2*c^2 + 8*a*b*c*d + 3*a^2*d^2 + 2*(b^2*c*d + 7*a*b
*d^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*
a^2*b*c*d^2 - a^3*d^3)*log(-4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x + a)*sq
rt(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*d^2), 1/48*(2*(8*b^2*d^2*x^2 - 3*b^2*c^2 + 8*a*
b*c*d + 3*a^2*d^2 + 2*(b^2*c*d + 7*a*b*d^2)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x
+ c) + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*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*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((b*x+a)**(3/2)*(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.250165, size = 456, normalized size = 2.96 \[ \frac{\frac{20 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b^{4} d^{2}} + \frac{b c d - a d^{2}}{b^{4} d^{4}}\right )} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{3} d^{3}}\right )} a{\left | b \right |}}{b^{2}} + \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{6} d^{2}} + \frac{b c d^{3} - 7 \, a d^{4}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{5} d^{4}}\right )}{\left | b \right |}}{b^{2}}}{1920 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*(20*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)/(b^4*
d^2) + (b*c*d - a*d^2)/(b^4*d^4)) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*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^3*d^3))
*a*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) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^5*d^4))*
abs(b)/b^2)/b