3.495 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^{7/2}} \, dx\)

Optimal. Leaf size=150 \[ b^{3/2} (5 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )+\frac{b^2 \sqrt{x} \sqrt{a+b x} (5 a B+2 A b)}{a}-\frac{2 (a+b x)^{5/2} (5 a B+2 A b)}{15 a x^{3/2}}-\frac{2 b (a+b x)^{3/2} (5 a B+2 A b)}{3 a \sqrt{x}}-\frac{2 A (a+b x)^{7/2}}{5 a x^{5/2}} \]

[Out]

(b^2*(2*A*b + 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/a - (2*b*(2*A*b + 5*a*B)*(a + b*x)^(
3/2))/(3*a*Sqrt[x]) - (2*(2*A*b + 5*a*B)*(a + b*x)^(5/2))/(15*a*x^(3/2)) - (2*A*
(a + b*x)^(7/2))/(5*a*x^(5/2)) + b^(3/2)*(2*A*b + 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x
])/Sqrt[a + b*x]]

_______________________________________________________________________________________

Rubi [A]  time = 0.171194, antiderivative size = 150, 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 \[ b^{3/2} (5 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )+\frac{b^2 \sqrt{x} \sqrt{a+b x} (5 a B+2 A b)}{a}-\frac{2 (a+b x)^{5/2} (5 a B+2 A b)}{15 a x^{3/2}}-\frac{2 b (a+b x)^{3/2} (5 a B+2 A b)}{3 a \sqrt{x}}-\frac{2 A (a+b x)^{7/2}}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/x^(7/2),x]

[Out]

(b^2*(2*A*b + 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/a - (2*b*(2*A*b + 5*a*B)*(a + b*x)^(
3/2))/(3*a*Sqrt[x]) - (2*(2*A*b + 5*a*B)*(a + b*x)^(5/2))/(15*a*x^(3/2)) - (2*A*
(a + b*x)^(7/2))/(5*a*x^(5/2)) + b^(3/2)*(2*A*b + 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x
])/Sqrt[a + b*x]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.3206, size = 146, normalized size = 0.97 \[ - \frac{2 A \left (a + b x\right )^{\frac{7}{2}}}{5 a x^{\frac{5}{2}}} + 2 b^{\frac{3}{2}} \left (A b + \frac{5 B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 b^{2} \sqrt{x} \sqrt{a + b x} \left (A b + \frac{5 B a}{2}\right )}{a} - \frac{2 b \left (a + b x\right )^{\frac{3}{2}} \left (2 A b + 5 B a\right )}{3 a \sqrt{x}} - \frac{4 \left (a + b x\right )^{\frac{5}{2}} \left (A b + \frac{5 B a}{2}\right )}{15 a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**(7/2),x)

[Out]

-2*A*(a + b*x)**(7/2)/(5*a*x**(5/2)) + 2*b**(3/2)*(A*b + 5*B*a/2)*atanh(sqrt(a +
b*x)/(sqrt(b)*sqrt(x))) + 2*b**2*sqrt(x)*sqrt(a + b*x)*(A*b + 5*B*a/2)/a - 2*b*
(a + b*x)**(3/2)*(2*A*b + 5*B*a)/(3*a*sqrt(x)) - 4*(a + b*x)**(5/2)*(A*b + 5*B*a
/2)/(15*a*x**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.175857, size = 100, normalized size = 0.67 \[ b^{3/2} (5 a B+2 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-\frac{\sqrt{a+b x} \left (2 a^2 (3 A+5 B x)+2 a b x (11 A+35 B x)+b^2 x^2 (46 A-15 B x)\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(7/2),x]

[Out]

-(Sqrt[a + b*x]*(b^2*x^2*(46*A - 15*B*x) + 2*a^2*(3*A + 5*B*x) + 2*a*b*x*(11*A +
35*B*x)))/(15*x^(5/2)) + b^(3/2)*(2*A*b + 5*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a
+ b*x]]

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 193, normalized size = 1.3 \[{\frac{1}{30}\sqrt{bx+a} \left ( 30\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{3}{b}^{5/2}+75\,B{b}^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) a{x}^{3}+30\,{b}^{2}B{x}^{3}\sqrt{x \left ( bx+a \right ) }-92\,A{x}^{2}{b}^{2}\sqrt{x \left ( bx+a \right ) }-140\,B{x}^{2}ab\sqrt{x \left ( bx+a \right ) }-44\,Axab\sqrt{x \left ( bx+a \right ) }-20\,Bx{a}^{2}\sqrt{x \left ( bx+a \right ) }-12\,A{a}^{2}\sqrt{x \left ( bx+a \right ) } \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^(7/2),x)

[Out]

1/30*(b*x+a)^(1/2)/x^(5/2)*(30*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^
(1/2))*x^3*b^(5/2)+75*B*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^3+30*b^2*B*x^3*(x*(b*x+a))^(1/2)-92*A*x^2*b^2*(x*(b*x+a))^(1/2)-140*B*
x^2*a*b*(x*(b*x+a))^(1/2)-44*A*x*a*b*(x*(b*x+a))^(1/2)-20*B*x*a^2*(x*(b*x+a))^(1
/2)-12*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^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.238385, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (5 \, B a b + 2 \, A b^{2}\right )} \sqrt{b} x^{3} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (15 \, B b^{2} x^{3} - 6 \, A a^{2} - 2 \,{\left (35 \, B a b + 23 \, A b^{2}\right )} x^{2} - 2 \,{\left (5 \, B a^{2} + 11 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{30 \, x^{3}}, \frac{15 \,{\left (5 \, B a b + 2 \, A b^{2}\right )} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) +{\left (15 \, B b^{2} x^{3} - 6 \, A a^{2} - 2 \,{\left (35 \, B a b + 23 \, A b^{2}\right )} x^{2} - 2 \,{\left (5 \, B a^{2} + 11 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{15 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(7/2),x, algorithm="fricas")

[Out]

[1/30*(15*(5*B*a*b + 2*A*b^2)*sqrt(b)*x^3*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sq
rt(x) + a) + 2*(15*B*b^2*x^3 - 6*A*a^2 - 2*(35*B*a*b + 23*A*b^2)*x^2 - 2*(5*B*a^
2 + 11*A*a*b)*x)*sqrt(b*x + a)*sqrt(x))/x^3, 1/15*(15*(5*B*a*b + 2*A*b^2)*sqrt(-
b)*x^3*arctan(sqrt(b*x + a)/(sqrt(-b)*sqrt(x))) + (15*B*b^2*x^3 - 6*A*a^2 - 2*(3
5*B*a*b + 23*A*b^2)*x^2 - 2*(5*B*a^2 + 11*A*a*b)*x)*sqrt(b*x + a)*sqrt(x))/x^3]

_______________________________________________________________________________________

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**(7/2),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^(7/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError