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

Optimal. Leaf size=144 \[ -\frac{(a+b x)^{3/2} \sqrt{c+d x}}{x}+2 b \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{a} (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}+\frac{\sqrt{b} (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.412674, antiderivative size = 144, 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 \[ -\frac{(a+b x)^{3/2} \sqrt{c+d x}}{x}+2 b \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{a} (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}+\frac{\sqrt{b} (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.3043, size = 133, normalized size = 0.92 \[ - \frac{\sqrt{a} \left (a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{c}} + \frac{\sqrt{b} \left (3 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{d}} + 2 b \sqrt{a + b x} \sqrt{c + d x} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.392265, size = 180, normalized size = 1.25 \[ \frac{1}{2} \left (2 \sqrt{a+b x} \left (b-\frac{a}{x}\right ) \sqrt{c+d x}+\frac{\sqrt{a} \log (x) (a d+3 b c)}{\sqrt{c}}-\frac{\sqrt{a} (a d+3 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}+\frac{\sqrt{b} (3 a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{d}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(b - a/x)*Sqrt[a + b*x]*Sqrt[c + d*x] + (Sqrt[a]*(3*b*c + a*d)*Log[x])/Sqrt[c
] - (Sqrt[a]*(3*b*c + a*d)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a
+ b*x]*Sqrt[c + d*x]])/Sqrt[c] + (Sqrt[b]*(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]])/Sqrt[d])/2

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Maple [B]  time = 0.022, size = 347, normalized size = 2.4 \[{\frac{1}{2\,x}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xabd\sqrt{ac}+\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) x{b}^{2}c\sqrt{ac}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ) x{a}^{2}d\sqrt{bd}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) xabc\sqrt{bd}+2\,xb\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-2\,a\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.84961, 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^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.602006, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x