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

Optimal. Leaf size=171 \[ -\frac{\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a} c^{3/2}}+2 b^{3/2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{4 c x} \]

[Out]

-((3*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*c*x) - ((a + b*x)^(3/2)*Sqrt[c +
d*x])/(2*x^2) - ((3*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*
x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*Sqrt[a]*c^(3/2)) + 2*b^(3/2)*Sqrt[d]*ArcTanh[(S
qrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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Rubi [A]  time = 0.419658, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a} c^{3/2}}+2 b^{3/2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{4 c x} \]

Antiderivative was successfully verified.

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

[Out]

-((3*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*c*x) - ((a + b*x)^(3/2)*Sqrt[c +
d*x])/(2*x^2) - ((3*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*
x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*Sqrt[a]*c^(3/2)) + 2*b^(3/2)*Sqrt[d]*ArcTanh[(S
qrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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Rubi in Sympy [A]  time = 65.7408, size = 156, normalized size = 0.91 \[ 2 b^{\frac{3}{2}} \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{2 x^{2}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d + 3 b c\right )}{4 c x} + \frac{\left (a^{2} d^{2} - 6 a b c d - 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 \sqrt{a} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**(3/2)*sqrt(d)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x))) - (a + b
*x)**(3/2)*sqrt(c + d*x)/(2*x**2) - sqrt(a + b*x)*sqrt(c + d*x)*(a*d + 3*b*c)/(4
*c*x) + (a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(
a)*sqrt(c + d*x)))/(4*sqrt(a)*c**(3/2))

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Mathematica [A]  time = 0.294842, size = 220, normalized size = 1.29 \[ -\frac{\log (x) \left (a^2 d^2-6 a b c d-3 b^2 c^2\right )}{8 \sqrt{a} c^{3/2}}+\frac{\left (a^2 d^2-6 a b c d-3 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 \sqrt{a} c^{3/2}}+b^{3/2} \sqrt{d} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )+\sqrt{a+b x} \sqrt{c+d x} \left (\frac{-a d-5 b c}{4 c x}-\frac{a}{2 x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-a/(2*x^2) + (-5*b*c - a*d)/(4*c*x))*Sqrt[a + b*x]*Sqrt[c + d*x] - ((-3*b^2*c^2
- 6*a*b*c*d + a^2*d^2)*Log[x])/(8*Sqrt[a]*c^(3/2)) + ((-3*b^2*c^2 - 6*a*b*c*d +
a^2*d^2)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d
*x]])/(8*Sqrt[a]*c^(3/2)) + b^(3/2)*Sqrt[d]*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*
Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]]

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Maple [B]  time = 0.022, size = 400, normalized size = 2.3 \[{\frac{1}{8\,c{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ){x}^{2}{a}^{2}{d}^{2}\sqrt{bd}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}abcd\sqrt{bd}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{2}\sqrt{bd}+8\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{2}cd\sqrt{ac}-2\,\sqrt{d{x}^{2}b+adx+bcx+ac}dax\sqrt{ac}\sqrt{bd}-10\,\sqrt{d{x}^{2}b+adx+bcx+ac}bxc\sqrt{ac}\sqrt{bd}-4\,\sqrt{d{x}^{2}b+adx+bcx+ac}ac\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c*(ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)+2*a*c)/x)*x^2*a^2*d^2*(b*d)^(1/2)-6*ln((a*d*x+b*c*x+2*(a*c)^(1/
2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^2*a*b*c*d*(b*d)^(1/2)-3*ln((a*d*x
+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^2*b^2*c^2*(b*d)
^(1/2)+8*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))*x^2*b^2*c*d*(a*c)^(1/2)-2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*a*x*(a*
c)^(1/2)*(b*d)^(1/2)-10*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*x*c*(a*c)^(1/2)*(b*d)^
(1/2)-4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*c*(a*c)^(1/2)*(b*d)^(1/2))/(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)/x^2/(a*c)^(1/2)/(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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.88495, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(8*sqrt(a*c)*sqrt(b*d)*b*c*x^2*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a
^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*
c*d + a*b*d^2)*x) - (3*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*x^2*log((4*(2*a^2*c^2 + (a
*b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c) + (8*a^2*c^2 + (b^2*c^2 + 6*a*b
*c*d + a^2*d^2)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c))/x^2) - 4*(2*a*c + (5*b
*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*c*x^2), 1/16*(16*
sqrt(a*c)*sqrt(-b*d)*b*c*x^2*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(-b*d)*sqrt(b
*x + a)*sqrt(d*x + c))) - (3*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*x^2*log((4*(2*a^2*c^
2 + (a*b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c) + (8*a^2*c^2 + (b^2*c^2 +
6*a*b*c*d + a^2*d^2)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c))/x^2) - 4*(2*a*c
+ (5*b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*c*x^2), 1/8
*(4*sqrt(-a*c)*sqrt(b*d)*b*c*x^2*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d
^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d
+ a*b*d^2)*x) - (3*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*x^2*arctan(1/2*(2*a*c + (b*c +
a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)) - 2*(2*a*c + (5*b*c + a*d
)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*c*x^2), 1/8*(8*sqrt(-a*
c)*sqrt(-b*d)*b*c*x^2*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(-b*d)*sqrt(b*x + a)
*sqrt(d*x + c))) - (3*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*x^2*arctan(1/2*(2*a*c + (b*
c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)) - 2*(2*a*c + (5*b*c +
a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*c*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.65473, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x