3.481 \(\int \frac{(a+b x)^{3/2} (A+B x)}{\sqrt{x}} \, dx\)

Optimal. Leaf size=126 \[ \frac{a^2 (6 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{3/2}}+\frac{\sqrt{x} (a+b x)^{3/2} (6 A b-a B)}{12 b}+\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-a B)}{8 b}+\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b} \]

[Out]

(a*(6*A*b - a*B)*Sqrt[x]*Sqrt[a + b*x])/(8*b) + ((6*A*b - a*B)*Sqrt[x]*(a + b*x)
^(3/2))/(12*b) + (B*Sqrt[x]*(a + b*x)^(5/2))/(3*b) + (a^2*(6*A*b - a*B)*ArcTanh[
(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*b^(3/2))

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Rubi [A]  time = 0.139065, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{a^2 (6 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{3/2}}+\frac{\sqrt{x} (a+b x)^{3/2} (6 A b-a B)}{12 b}+\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-a B)}{8 b}+\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

(a*(6*A*b - a*B)*Sqrt[x]*Sqrt[a + b*x])/(8*b) + ((6*A*b - a*B)*Sqrt[x]*(a + b*x)
^(3/2))/(12*b) + (B*Sqrt[x]*(a + b*x)^(5/2))/(3*b) + (a^2*(6*A*b - a*B)*ArcTanh[
(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*b^(3/2))

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Rubi in Sympy [A]  time = 11.7932, size = 109, normalized size = 0.87 \[ \frac{B \sqrt{x} \left (a + b x\right )^{\frac{5}{2}}}{3 b} + \frac{a^{2} \left (6 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{3}{2}}} + \frac{a \sqrt{x} \sqrt{a + b x} \left (6 A b - B a\right )}{8 b} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (6 A b - B a\right )}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*sqrt(x)*(a + b*x)**(5/2)/(3*b) + a**2*(6*A*b - B*a)*atanh(sqrt(a + b*x)/(sqrt(
b)*sqrt(x)))/(8*b**(3/2)) + a*sqrt(x)*sqrt(a + b*x)*(6*A*b - B*a)/(8*b) + sqrt(x
)*(a + b*x)**(3/2)*(6*A*b - B*a)/(12*b)

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Mathematica [A]  time = 0.0990434, size = 100, normalized size = 0.79 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (3 a^2 B+2 a b (15 A+7 B x)+4 b^2 x (3 A+2 B x)\right )-3 a^2 (a B-6 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{24 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(3*a^2*B + 4*b^2*x*(3*A + 2*B*x) + 2*a*b*(15*A +
7*B*x)) - 3*a^2*(-6*A*b + a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(24*b^(3/
2))

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Maple [A]  time = 0.017, size = 176, normalized size = 1.4 \[{\frac{1}{48}\sqrt{bx+a}\sqrt{x} \left ( 16\,B{x}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+24\,A\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}+28\,Ba\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+18\,A{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b+60\,A\sqrt{x \left ( bx+a \right ) }a{b}^{3/2}-3\,B{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +6\,B{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(b*x+a)^(1/2)*x^(1/2)/b^(3/2)*(16*B*x^2*b^(5/2)*(x*(b*x+a))^(1/2)+24*A*(x*(
b*x+a))^(1/2)*x*b^(5/2)+28*B*a*(x*(b*x+a))^(1/2)*x*b^(3/2)+18*A*a^2*ln(1/2*(2*(x
*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b+60*A*(x*(b*x+a))^(1/2)*a*b^(3/2)-3*B
*a^3*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+6*B*a^2*(x*(b*x+a))^(
1/2)*b^(1/2))/(x*(b*x+a))^(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)*(b*x + a)^(3/2)/sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247875, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{2} + 3 \, B a^{2} + 30 \, A a b + 2 \,{\left (7 \, B a b + 6 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 3 \,{\left (B a^{3} - 6 \, A a^{2} b\right )} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{48 \, b^{\frac{3}{2}}}, \frac{{\left (8 \, B b^{2} x^{2} + 3 \, B a^{2} + 30 \, A a b + 2 \,{\left (7 \, B a b + 6 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} - 3 \,{\left (B a^{3} - 6 \, A a^{2} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{24 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(2*(8*B*b^2*x^2 + 3*B*a^2 + 30*A*a*b + 2*(7*B*a*b + 6*A*b^2)*x)*sqrt(b*x +
a)*sqrt(b)*sqrt(x) - 3*(B*a^3 - 6*A*a^2*b)*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b
*x + a)*sqrt(b)))/b^(3/2), 1/24*((8*B*b^2*x^2 + 3*B*a^2 + 30*A*a*b + 2*(7*B*a*b
+ 6*A*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) - 3*(B*a^3 - 6*A*a^2*b)*arctan(sqrt
(b*x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b)]

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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)*(B*x+A)/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError