3.478 \(\int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx\)

Optimal. Leaf size=225 \[ -\frac{a^5 (12 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^{9/2}}+\frac{a^4 \sqrt{x} \sqrt{a+b x} (12 A b-7 a B)}{512 b^4}-\frac{a^3 x^{3/2} \sqrt{a+b x} (12 A b-7 a B)}{768 b^3}+\frac{a^2 x^{5/2} \sqrt{a+b x} (12 A b-7 a B)}{960 b^2}+\frac{a x^{7/2} \sqrt{a+b x} (12 A b-7 a B)}{160 b}+\frac{x^{7/2} (a+b x)^{3/2} (12 A b-7 a B)}{60 b}+\frac{B x^{7/2} (a+b x)^{5/2}}{6 b} \]

[Out]

(a^4*(12*A*b - 7*a*B)*Sqrt[x]*Sqrt[a + b*x])/(512*b^4) - (a^3*(12*A*b - 7*a*B)*x
^(3/2)*Sqrt[a + b*x])/(768*b^3) + (a^2*(12*A*b - 7*a*B)*x^(5/2)*Sqrt[a + b*x])/(
960*b^2) + (a*(12*A*b - 7*a*B)*x^(7/2)*Sqrt[a + b*x])/(160*b) + ((12*A*b - 7*a*B
)*x^(7/2)*(a + b*x)^(3/2))/(60*b) + (B*x^(7/2)*(a + b*x)^(5/2))/(6*b) - (a^5*(12
*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(512*b^(9/2))

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Rubi [A]  time = 0.285282, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^5 (12 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^{9/2}}+\frac{a^4 \sqrt{x} \sqrt{a+b x} (12 A b-7 a B)}{512 b^4}-\frac{a^3 x^{3/2} \sqrt{a+b x} (12 A b-7 a B)}{768 b^3}+\frac{a^2 x^{5/2} \sqrt{a+b x} (12 A b-7 a B)}{960 b^2}+\frac{a x^{7/2} \sqrt{a+b x} (12 A b-7 a B)}{160 b}+\frac{x^{7/2} (a+b x)^{3/2} (12 A b-7 a B)}{60 b}+\frac{B x^{7/2} (a+b x)^{5/2}}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a + b*x)^(3/2)*(A + B*x),x]

[Out]

(a^4*(12*A*b - 7*a*B)*Sqrt[x]*Sqrt[a + b*x])/(512*b^4) - (a^3*(12*A*b - 7*a*B)*x
^(3/2)*Sqrt[a + b*x])/(768*b^3) + (a^2*(12*A*b - 7*a*B)*x^(5/2)*Sqrt[a + b*x])/(
960*b^2) + (a*(12*A*b - 7*a*B)*x^(7/2)*Sqrt[a + b*x])/(160*b) + ((12*A*b - 7*a*B
)*x^(7/2)*(a + b*x)^(3/2))/(60*b) + (B*x^(7/2)*(a + b*x)^(5/2))/(6*b) - (a^5*(12
*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(512*b^(9/2))

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Rubi in Sympy [A]  time = 26.7556, size = 214, normalized size = 0.95 \[ \frac{B x^{\frac{7}{2}} \left (a + b x\right )^{\frac{5}{2}}}{6 b} - \frac{a^{5} \left (12 A b - 7 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{512 b^{\frac{9}{2}}} + \frac{a^{4} \sqrt{x} \sqrt{a + b x} \left (12 A b - 7 B a\right )}{512 b^{4}} + \frac{a^{3} x^{\frac{3}{2}} \sqrt{a + b x} \left (12 A b - 7 B a\right )}{256 b^{3}} + \frac{a^{2} x^{\frac{3}{2}} \left (a + b x\right )^{\frac{3}{2}} \left (12 A b - 7 B a\right )}{192 b^{3}} - \frac{a x^{\frac{3}{2}} \left (a + b x\right )^{\frac{5}{2}} \left (12 A b - 7 B a\right )}{96 b^{3}} + \frac{x^{\frac{5}{2}} \left (a + b x\right )^{\frac{5}{2}} \left (12 A b - 7 B a\right )}{60 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(b*x+a)**(3/2)*(B*x+A),x)

[Out]

B*x**(7/2)*(a + b*x)**(5/2)/(6*b) - a**5*(12*A*b - 7*B*a)*atanh(sqrt(b)*sqrt(x)/
sqrt(a + b*x))/(512*b**(9/2)) + a**4*sqrt(x)*sqrt(a + b*x)*(12*A*b - 7*B*a)/(512
*b**4) + a**3*x**(3/2)*sqrt(a + b*x)*(12*A*b - 7*B*a)/(256*b**3) + a**2*x**(3/2)
*(a + b*x)**(3/2)*(12*A*b - 7*B*a)/(192*b**3) - a*x**(3/2)*(a + b*x)**(5/2)*(12*
A*b - 7*B*a)/(96*b**3) + x**(5/2)*(a + b*x)**(5/2)*(12*A*b - 7*B*a)/(60*b**2)

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Mathematica [A]  time = 0.196802, size = 157, normalized size = 0.7 \[ \frac{15 a^5 (7 a B-12 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )+\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (-105 a^5 B+10 a^4 b (18 A+7 B x)-8 a^3 b^2 x (15 A+7 B x)+48 a^2 b^3 x^2 (2 A+B x)+64 a b^4 x^3 (33 A+26 B x)+256 b^5 x^4 (6 A+5 B x)\right )}{7680 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*(a + b*x)^(3/2)*(A + B*x),x]

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(-105*a^5*B + 48*a^2*b^3*x^2*(2*A + B*x) + 256*b^
5*x^4*(6*A + 5*B*x) - 8*a^3*b^2*x*(15*A + 7*B*x) + 10*a^4*b*(18*A + 7*B*x) + 64*
a*b^4*x^3*(33*A + 26*B*x)) + 15*a^5*(-12*A*b + 7*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sq
rt[a + b*x]])/(7680*b^(9/2))

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Maple [A]  time = 0.02, size = 302, normalized size = 1.3 \[ -{\frac{1}{15360}\sqrt{x}\sqrt{bx+a} \left ( -2560\,B{x}^{5}{b}^{11/2}\sqrt{x \left ( bx+a \right ) }-3072\,A{x}^{4}{b}^{11/2}\sqrt{x \left ( bx+a \right ) }-3328\,B{x}^{4}a{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-4224\,A{x}^{3}a{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-96\,B{x}^{3}{a}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-192\,A{x}^{2}{a}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+112\,B{x}^{2}{a}^{3}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+240\,{a}^{3}\sqrt{x \left ( bx+a \right ) }xA{b}^{5/2}-140\,{a}^{4}\sqrt{x \left ( bx+a \right ) }xB{b}^{3/2}+180\,{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) Ab-360\,{a}^{4}\sqrt{x \left ( bx+a \right ) }A{b}^{3/2}-105\,{a}^{6}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) B+210\,{a}^{5}\sqrt{x \left ( bx+a \right ) }B\sqrt{b} \right ){b}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x+a)^(3/2)*(B*x+A),x)

[Out]

-1/15360*x^(1/2)*(b*x+a)^(1/2)/b^(9/2)*(-2560*B*x^5*b^(11/2)*(x*(b*x+a))^(1/2)-3
072*A*x^4*b^(11/2)*(x*(b*x+a))^(1/2)-3328*B*x^4*a*b^(9/2)*(x*(b*x+a))^(1/2)-4224
*A*x^3*a*b^(9/2)*(x*(b*x+a))^(1/2)-96*B*x^3*a^2*b^(7/2)*(x*(b*x+a))^(1/2)-192*A*
x^2*a^2*b^(7/2)*(x*(b*x+a))^(1/2)+112*B*x^2*a^3*b^(5/2)*(x*(b*x+a))^(1/2)+240*a^
3*(x*(b*x+a))^(1/2)*x*A*b^(5/2)-140*a^4*(x*(b*x+a))^(1/2)*x*B*b^(3/2)+180*a^5*ln
(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*A*b-360*a^4*(x*(b*x+a))^(1/2
)*A*b^(3/2)-105*a^6*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*B+210*
a^5*(x*(b*x+a))^(1/2)*B*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)*x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, B b^{5} x^{5} - 105 \, B a^{5} + 180 \, A a^{4} b + 128 \,{\left (13 \, B a b^{4} + 12 \, A b^{5}\right )} x^{4} + 48 \,{\left (B a^{2} b^{3} + 44 \, A a b^{4}\right )} x^{3} - 8 \,{\left (7 \, B a^{3} b^{2} - 12 \, A a^{2} b^{3}\right )} x^{2} + 10 \,{\left (7 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 15 \,{\left (7 \, B a^{6} - 12 \, A a^{5} b\right )} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{15360 \, b^{\frac{9}{2}}}, \frac{{\left (1280 \, B b^{5} x^{5} - 105 \, B a^{5} + 180 \, A a^{4} b + 128 \,{\left (13 \, B a b^{4} + 12 \, A b^{5}\right )} x^{4} + 48 \,{\left (B a^{2} b^{3} + 44 \, A a b^{4}\right )} x^{3} - 8 \,{\left (7 \, B a^{3} b^{2} - 12 \, A a^{2} b^{3}\right )} x^{2} + 10 \,{\left (7 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} + 15 \,{\left (7 \, B a^{6} - 12 \, A a^{5} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{7680 \, \sqrt{-b} b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)*x^(5/2),x, algorithm="fricas")

[Out]

[1/15360*(2*(1280*B*b^5*x^5 - 105*B*a^5 + 180*A*a^4*b + 128*(13*B*a*b^4 + 12*A*b
^5)*x^4 + 48*(B*a^2*b^3 + 44*A*a*b^4)*x^3 - 8*(7*B*a^3*b^2 - 12*A*a^2*b^3)*x^2 +
10*(7*B*a^4*b - 12*A*a^3*b^2)*x)*sqrt(b*x + a)*sqrt(b)*sqrt(x) - 15*(7*B*a^6 -
12*A*a^5*b)*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)))/b^(9/2), 1/76
80*((1280*B*b^5*x^5 - 105*B*a^5 + 180*A*a^4*b + 128*(13*B*a*b^4 + 12*A*b^5)*x^4
+ 48*(B*a^2*b^3 + 44*A*a*b^4)*x^3 - 8*(7*B*a^3*b^2 - 12*A*a^2*b^3)*x^2 + 10*(7*B
*a^4*b - 12*A*a^3*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) + 15*(7*B*a^6 - 12*A*a^
5*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)**(3/2)*(B*x+A),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)*(b*x + a)^(3/2)*x^(5/2),x, algorithm="giac")

[Out]

Timed out