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

Optimal. Leaf size=192 \[ -\frac{a^4 (10 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{9/2}}+\frac{a^3 \sqrt{x} \sqrt{a+b x} (10 A b-7 a B)}{128 b^4}-\frac{a^2 x^{3/2} \sqrt{a+b x} (10 A b-7 a B)}{192 b^3}+\frac{a x^{5/2} \sqrt{a+b x} (10 A b-7 a B)}{240 b^2}+\frac{x^{7/2} \sqrt{a+b x} (10 A b-7 a B)}{40 b}+\frac{B x^{7/2} (a+b x)^{3/2}}{5 b} \]

[Out]

(a^3*(10*A*b - 7*a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^4) - (a^2*(10*A*b - 7*a*B)*x
^(3/2)*Sqrt[a + b*x])/(192*b^3) + (a*(10*A*b - 7*a*B)*x^(5/2)*Sqrt[a + b*x])/(24
0*b^2) + ((10*A*b - 7*a*B)*x^(7/2)*Sqrt[a + b*x])/(40*b) + (B*x^(7/2)*(a + b*x)^
(3/2))/(5*b) - (a^4*(10*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(
128*b^(9/2))

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Rubi [A]  time = 0.235583, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^4 (10 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{9/2}}+\frac{a^3 \sqrt{x} \sqrt{a+b x} (10 A b-7 a B)}{128 b^4}-\frac{a^2 x^{3/2} \sqrt{a+b x} (10 A b-7 a B)}{192 b^3}+\frac{a x^{5/2} \sqrt{a+b x} (10 A b-7 a B)}{240 b^2}+\frac{x^{7/2} \sqrt{a+b x} (10 A b-7 a B)}{40 b}+\frac{B x^{7/2} (a+b x)^{3/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*(10*A*b - 7*a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^4) - (a^2*(10*A*b - 7*a*B)*x
^(3/2)*Sqrt[a + b*x])/(192*b^3) + (a*(10*A*b - 7*a*B)*x^(5/2)*Sqrt[a + b*x])/(24
0*b^2) + ((10*A*b - 7*a*B)*x^(7/2)*Sqrt[a + b*x])/(40*b) + (B*x^(7/2)*(a + b*x)^
(3/2))/(5*b) - (a^4*(10*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(
128*b^(9/2))

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Rubi in Sympy [A]  time = 21.6061, size = 182, normalized size = 0.95 \[ \frac{B x^{\frac{7}{2}} \left (a + b x\right )^{\frac{3}{2}}}{5 b} - \frac{a^{4} \left (10 A b - 7 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{128 b^{\frac{9}{2}}} + \frac{a^{3} \sqrt{x} \sqrt{a + b x} \left (10 A b - 7 B a\right )}{128 b^{4}} + \frac{a^{2} x^{\frac{3}{2}} \sqrt{a + b x} \left (10 A b - 7 B a\right )}{64 b^{3}} - \frac{a x^{\frac{3}{2}} \left (a + b x\right )^{\frac{3}{2}} \left (10 A b - 7 B a\right )}{48 b^{3}} + \frac{x^{\frac{5}{2}} \left (a + b x\right )^{\frac{3}{2}} \left (10 A b - 7 B a\right )}{40 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**(7/2)*(a + b*x)**(3/2)/(5*b) - a**4*(10*A*b - 7*B*a)*atanh(sqrt(a + b*x)/(s
qrt(b)*sqrt(x)))/(128*b**(9/2)) + a**3*sqrt(x)*sqrt(a + b*x)*(10*A*b - 7*B*a)/(1
28*b**4) + a**2*x**(3/2)*sqrt(a + b*x)*(10*A*b - 7*B*a)/(64*b**3) - a*x**(3/2)*(
a + b*x)**(3/2)*(10*A*b - 7*B*a)/(48*b**3) + x**(5/2)*(a + b*x)**(3/2)*(10*A*b -
7*B*a)/(40*b**2)

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Mathematica [A]  time = 0.165992, size = 139, normalized size = 0.72 \[ \frac{15 a^4 (7 a B-10 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )+\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (-105 a^4 B+10 a^3 b (15 A+7 B x)-4 a^2 b^2 x (25 A+14 B x)+16 a b^3 x^2 (5 A+3 B x)+96 b^4 x^3 (5 A+4 B x)\right )}{1920 b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(-105*a^4*B + 16*a*b^3*x^2*(5*A + 3*B*x) + 96*b^4
*x^3*(5*A + 4*B*x) + 10*a^3*b*(15*A + 7*B*x) - 4*a^2*b^2*x*(25*A + 14*B*x)) + 15
*a^4*(-10*A*b + 7*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(1920*b^(9/2))

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Maple [A]  time = 0.021, size = 260, normalized size = 1.4 \[ -{\frac{1}{3840}\sqrt{x}\sqrt{bx+a} \left ( -768\,B{x}^{4}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-960\,A{x}^{3}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-96\,B{x}^{3}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-160\,A{x}^{2}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+112\,B{x}^{2}{a}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+200\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}-140\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+150\,A{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-300\,A{a}^{3}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-105\,B{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +210\,B{a}^{4}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3840*x^(1/2)*(b*x+a)^(1/2)/b^(9/2)*(-768*B*x^4*b^(9/2)*(x*(b*x+a))^(1/2)-960*
A*x^3*b^(9/2)*(x*(b*x+a))^(1/2)-96*B*x^3*a*b^(7/2)*(x*(b*x+a))^(1/2)-160*A*x^2*a
*b^(7/2)*(x*(b*x+a))^(1/2)+112*B*x^2*a^2*b^(5/2)*(x*(b*x+a))^(1/2)+200*A*a^2*(x*
(b*x+a))^(1/2)*x*b^(5/2)-140*B*a^3*(x*(b*x+a))^(1/2)*x*b^(3/2)+150*A*a^4*ln(1/2*
(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-300*A*a^3*(x*(b*x+a))^(1/2)*b^(
3/2)-105*B*a^5*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+210*B*a^4*(
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)*sqrt(b*x + a)*x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245727, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (384 \, B b^{4} x^{4} - 105 \, B a^{4} + 150 \, A a^{3} b + 48 \,{\left (B a b^{3} + 10 \, A b^{4}\right )} x^{3} - 8 \,{\left (7 \, B a^{2} b^{2} - 10 \, A a b^{3}\right )} x^{2} + 10 \,{\left (7 \, B a^{3} b - 10 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 15 \,{\left (7 \, B a^{5} - 10 \, A a^{4} b\right )} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{3840 \, b^{\frac{9}{2}}}, \frac{{\left (384 \, B b^{4} x^{4} - 105 \, B a^{4} + 150 \, A a^{3} b + 48 \,{\left (B a b^{3} + 10 \, A b^{4}\right )} x^{3} - 8 \,{\left (7 \, B a^{2} b^{2} - 10 \, A a b^{3}\right )} x^{2} + 10 \,{\left (7 \, B a^{3} b - 10 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} + 15 \,{\left (7 \, B a^{5} - 10 \, A a^{4} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{1920 \, \sqrt{-b} b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3840*(2*(384*B*b^4*x^4 - 105*B*a^4 + 150*A*a^3*b + 48*(B*a*b^3 + 10*A*b^4)*x^
3 - 8*(7*B*a^2*b^2 - 10*A*a*b^3)*x^2 + 10*(7*B*a^3*b - 10*A*a^2*b^2)*x)*sqrt(b*x
+ a)*sqrt(b)*sqrt(x) - 15*(7*B*a^5 - 10*A*a^4*b)*log(-2*sqrt(b*x + a)*b*sqrt(x)
+ (2*b*x + a)*sqrt(b)))/b^(9/2), 1/1920*((384*B*b^4*x^4 - 105*B*a^4 + 150*A*a^3
*b + 48*(B*a*b^3 + 10*A*b^4)*x^3 - 8*(7*B*a^2*b^2 - 10*A*a*b^3)*x^2 + 10*(7*B*a^
3*b - 10*A*a^2*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) + 15*(7*B*a^5 - 10*A*a^4*b
)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^4)]

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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(x**(5/2)*(B*x+A)*(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [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)*sqrt(b*x + a)*x^(5/2),x, algorithm="giac")

[Out]

Timed out