3.492 \(\int \frac{(a+b x)^{5/2} (A+B x)}{\sqrt{x}} \, dx\)

Optimal. Leaf size=159 \[ \frac{5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{3/2}}+\frac{5 a^2 \sqrt{x} \sqrt{a+b x} (8 A b-a B)}{64 b}+\frac{\sqrt{x} (a+b x)^{5/2} (8 A b-a B)}{24 b}+\frac{5 a \sqrt{x} (a+b x)^{3/2} (8 A b-a B)}{96 b}+\frac{B \sqrt{x} (a+b x)^{7/2}}{4 b} \]

[Out]

(5*a^2*(8*A*b - a*B)*Sqrt[x]*Sqrt[a + b*x])/(64*b) + (5*a*(8*A*b - a*B)*Sqrt[x]*
(a + b*x)^(3/2))/(96*b) + ((8*A*b - a*B)*Sqrt[x]*(a + b*x)^(5/2))/(24*b) + (B*Sq
rt[x]*(a + b*x)^(7/2))/(4*b) + (5*a^3*(8*A*b - a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sq
rt[a + b*x]])/(64*b^(3/2))

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Rubi [A]  time = 0.175914, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{3/2}}+\frac{5 a^2 \sqrt{x} \sqrt{a+b x} (8 A b-a B)}{64 b}+\frac{\sqrt{x} (a+b x)^{5/2} (8 A b-a B)}{24 b}+\frac{5 a \sqrt{x} (a+b x)^{3/2} (8 A b-a B)}{96 b}+\frac{B \sqrt{x} (a+b x)^{7/2}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/Sqrt[x],x]

[Out]

(5*a^2*(8*A*b - a*B)*Sqrt[x]*Sqrt[a + b*x])/(64*b) + (5*a*(8*A*b - a*B)*Sqrt[x]*
(a + b*x)^(3/2))/(96*b) + ((8*A*b - a*B)*Sqrt[x]*(a + b*x)^(5/2))/(24*b) + (B*Sq
rt[x]*(a + b*x)^(7/2))/(4*b) + (5*a^3*(8*A*b - a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sq
rt[a + b*x]])/(64*b^(3/2))

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Rubi in Sympy [A]  time = 15.4212, size = 143, normalized size = 0.9 \[ \frac{B \sqrt{x} \left (a + b x\right )^{\frac{7}{2}}}{4 b} + \frac{5 a^{3} \left (8 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{64 b^{\frac{3}{2}}} + \frac{5 a^{2} \sqrt{x} \sqrt{a + b x} \left (8 A b - B a\right )}{64 b} + \frac{5 a \sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (8 A b - B a\right )}{96 b} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{5}{2}} \left (8 A b - B a\right )}{24 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**(1/2),x)

[Out]

B*sqrt(x)*(a + b*x)**(7/2)/(4*b) + 5*a**3*(8*A*b - B*a)*atanh(sqrt(b)*sqrt(x)/sq
rt(a + b*x))/(64*b**(3/2)) + 5*a**2*sqrt(x)*sqrt(a + b*x)*(8*A*b - B*a)/(64*b) +
5*a*sqrt(x)*(a + b*x)**(3/2)*(8*A*b - B*a)/(96*b) + sqrt(x)*(a + b*x)**(5/2)*(8
*A*b - B*a)/(24*b)

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Mathematica [A]  time = 0.137843, size = 119, normalized size = 0.75 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (15 a^3 B+2 a^2 b (132 A+59 B x)+8 a b^2 x (26 A+17 B x)+16 b^3 x^2 (4 A+3 B x)\right )-15 a^3 (a B-8 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{192 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/Sqrt[x],x]

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(15*a^3*B + 16*b^3*x^2*(4*A + 3*B*x) + 8*a*b^2*x*
(26*A + 17*B*x) + 2*a^2*b*(132*A + 59*B*x)) - 15*a^3*(-8*A*b + a*B)*Log[b*Sqrt[x
] + Sqrt[b]*Sqrt[a + b*x]])/(192*b^(3/2))

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Maple [A]  time = 0.019, size = 218, normalized size = 1.4 \[{\frac{1}{384}\sqrt{bx+a}\sqrt{x} \left ( 96\,B{x}^{3}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+128\,A{x}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+272\,B{x}^{2}a{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+416\,Aax\sqrt{x \left ( bx+a \right ) }{b}^{5/2}+236\,B{a}^{2}x\sqrt{x \left ( bx+a \right ) }{b}^{3/2}+120\,A{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b+528\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-15\,B{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +30\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^(1/2),x)

[Out]

1/384*(b*x+a)^(1/2)*x^(1/2)/b^(3/2)*(96*B*x^3*b^(7/2)*(x*(b*x+a))^(1/2)+128*A*x^
2*b^(7/2)*(x*(b*x+a))^(1/2)+272*B*x^2*a*b^(5/2)*(x*(b*x+a))^(1/2)+416*A*a*x*(x*(
b*x+a))^(1/2)*b^(5/2)+236*B*a^2*x*(x*(b*x+a))^(1/2)*b^(3/2)+120*A*a^3*ln(1/2*(2*
(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b+528*A*a^2*(x*(b*x+a))^(1/2)*b^(3/2
)-15*B*a^4*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+30*B*a^3*(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)^(5/2)/sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237794, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B b^{3} x^{3} + 15 \, B a^{3} + 264 \, A a^{2} b + 8 \,{\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{2} + 2 \,{\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{384 \, b^{\frac{3}{2}}}, \frac{{\left (48 \, B b^{3} x^{3} + 15 \, B a^{3} + 264 \, A a^{2} b + 8 \,{\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{2} + 2 \,{\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} - 15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{192 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/sqrt(x),x, algorithm="fricas")

[Out]

[1/384*(2*(48*B*b^3*x^3 + 15*B*a^3 + 264*A*a^2*b + 8*(17*B*a*b^2 + 8*A*b^3)*x^2
+ 2*(59*B*a^2*b + 104*A*a*b^2)*x)*sqrt(b*x + a)*sqrt(b)*sqrt(x) - 15*(B*a^4 - 8*
A*a^3*b)*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)))/b^(3/2), 1/192*((
48*B*b^3*x^3 + 15*B*a^3 + 264*A*a^2*b + 8*(17*B*a*b^2 + 8*A*b^3)*x^2 + 2*(59*B*a
^2*b + 104*A*a*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) - 15*(B*a^4 - 8*A*a^3*b)*a
rctan(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)**(5/2)*(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)^(5/2)/sqrt(x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError