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

Optimal. Leaf size=152 \[ -\frac{2 (a+b x)^{5/2} (3 a B+4 A b)}{3 a \sqrt{x}}+\frac{5 b \sqrt{x} (a+b x)^{3/2} (3 a B+4 A b)}{6 a}+\frac{5}{4} b \sqrt{x} \sqrt{a+b x} (3 a B+4 A b)+\frac{5}{4} a \sqrt{b} (3 a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 A (a+b x)^{7/2}}{3 a x^{3/2}} \]

[Out]

(5*b*(4*A*b + 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/4 + (5*b*(4*A*b + 3*a*B)*Sqrt[x]*(a
+ b*x)^(3/2))/(6*a) - (2*(4*A*b + 3*a*B)*(a + b*x)^(5/2))/(3*a*Sqrt[x]) - (2*A*(
a + b*x)^(7/2))/(3*a*x^(3/2)) + (5*a*Sqrt[b]*(4*A*b + 3*a*B)*ArcTanh[(Sqrt[b]*Sq
rt[x])/Sqrt[a + b*x]])/4

_______________________________________________________________________________________

Rubi [A]  time = 0.173046, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 (a+b x)^{5/2} (3 a B+4 A b)}{3 a \sqrt{x}}+\frac{5 b \sqrt{x} (a+b x)^{3/2} (3 a B+4 A b)}{6 a}+\frac{5}{4} b \sqrt{x} \sqrt{a+b x} (3 a B+4 A b)+\frac{5}{4} a \sqrt{b} (3 a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 A (a+b x)^{7/2}}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(5*b*(4*A*b + 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/4 + (5*b*(4*A*b + 3*a*B)*Sqrt[x]*(a
+ b*x)^(3/2))/(6*a) - (2*(4*A*b + 3*a*B)*(a + b*x)^(5/2))/(3*a*Sqrt[x]) - (2*A*(
a + b*x)^(7/2))/(3*a*x^(3/2)) + (5*a*Sqrt[b]*(4*A*b + 3*a*B)*ArcTanh[(Sqrt[b]*Sq
rt[x])/Sqrt[a + b*x]])/4

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.9188, size = 148, normalized size = 0.97 \[ - \frac{2 A \left (a + b x\right )^{\frac{7}{2}}}{3 a x^{\frac{3}{2}}} + \frac{5 a \sqrt{b} \left (4 A b + 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{4} + \frac{5 b \sqrt{x} \sqrt{a + b x} \left (4 A b + 3 B a\right )}{4} + \frac{5 b \sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (4 A b + 3 B a\right )}{6 a} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (4 A b + 3 B a\right )}{3 a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*(a + b*x)**(7/2)/(3*a*x**(3/2)) + 5*a*sqrt(b)*(4*A*b + 3*B*a)*atanh(sqrt(b)
*sqrt(x)/sqrt(a + b*x))/4 + 5*b*sqrt(x)*sqrt(a + b*x)*(4*A*b + 3*B*a)/4 + 5*b*sq
rt(x)*(a + b*x)**(3/2)*(4*A*b + 3*B*a)/(6*a) - 2*(a + b*x)**(5/2)*(4*A*b + 3*B*a
)/(3*a*sqrt(x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.164309, size = 101, normalized size = 0.66 \[ \frac{\sqrt{a+b x} \left (-8 a^2 (A+3 B x)+a b x (27 B x-56 A)+6 b^2 x^2 (2 A+B x)\right )}{12 x^{3/2}}+\frac{5}{4} a \sqrt{b} (3 a B+4 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(6*b^2*x^2*(2*A + B*x) - 8*a^2*(A + 3*B*x) + a*b*x*(-56*A + 27*B*
x)))/(12*x^(3/2)) + (5*a*Sqrt[b]*(4*A*b + 3*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a
+ b*x]])/4

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 196, normalized size = 1.3 \[{\frac{1}{24}\sqrt{bx+a} \left ( 60\,a{b}^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) A{x}^{2}+45\,B\sqrt{b}{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}+12\,{b}^{2}B{x}^{3}\sqrt{x \left ( bx+a \right ) }+24\,A{x}^{2}{b}^{2}\sqrt{x \left ( bx+a \right ) }+54\,B{x}^{2}ab\sqrt{x \left ( bx+a \right ) }-112\,Axab\sqrt{x \left ( bx+a \right ) }-48\,Bx{a}^{2}\sqrt{x \left ( bx+a \right ) }-16\,A{a}^{2}\sqrt{x \left ( bx+a \right ) } \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(b*x+a)^(1/2)/x^(3/2)*(60*a*b^(3/2)*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b
*x+a)/b^(1/2))*A*x^2+45*B*b^(1/2)*a^2*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+
a)/b^(1/2))*x^2+12*b^2*B*x^3*(x*(b*x+a))^(1/2)+24*A*x^2*b^2*(x*(b*x+a))^(1/2)+54
*B*x^2*a*b*(x*(b*x+a))^(1/2)-112*A*x*a*b*(x*(b*x+a))^(1/2)-48*B*x*a^2*(x*(b*x+a)
)^(1/2)-16*A*a^2*(x*(b*x+a))^(1/2))/(x*(b*x+a))^(1/2)

_______________________________________________________________________________________

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)/x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.228717, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt{b} x^{2} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (6 \, B b^{2} x^{3} - 8 \, A a^{2} + 3 \,{\left (9 \, B a b + 4 \, A b^{2}\right )} x^{2} - 8 \,{\left (3 \, B a^{2} + 7 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{24 \, x^{2}}, \frac{15 \,{\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) +{\left (6 \, B b^{2} x^{3} - 8 \, A a^{2} + 3 \,{\left (9 \, B a b + 4 \, A b^{2}\right )} x^{2} - 8 \,{\left (3 \, B a^{2} + 7 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{12 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/24*(15*(3*B*a^2 + 4*A*a*b)*sqrt(b)*x^2*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sq
rt(x) + a) + 2*(6*B*b^2*x^3 - 8*A*a^2 + 3*(9*B*a*b + 4*A*b^2)*x^2 - 8*(3*B*a^2 +
7*A*a*b)*x)*sqrt(b*x + a)*sqrt(x))/x^2, 1/12*(15*(3*B*a^2 + 4*A*a*b)*sqrt(-b)*x
^2*arctan(sqrt(b*x + a)/(sqrt(-b)*sqrt(x))) + (6*B*b^2*x^3 - 8*A*a^2 + 3*(9*B*a*
b + 4*A*b^2)*x^2 - 8*(3*B*a^2 + 7*A*a*b)*x)*sqrt(b*x + a)*sqrt(x))/x^2]

_______________________________________________________________________________________

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**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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)/x^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError