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

Optimal. Leaf size=200 \[ \frac{35 a^2 (2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{11/2}}-\frac{35 a \sqrt{x} \sqrt{a+b x} (2 A b-3 a B)}{8 b^5}+\frac{35 x^{3/2} \sqrt{a+b x} (2 A b-3 a B)}{12 b^4}-\frac{7 x^{5/2} \sqrt{a+b x} (2 A b-3 a B)}{3 a b^3}+\frac{2 x^{7/2} (2 A b-3 a B)}{a b^2 \sqrt{a+b x}}+\frac{2 x^{9/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

[Out]

(2*(A*b - a*B)*x^(9/2))/(3*a*b*(a + b*x)^(3/2)) + (2*(2*A*b - 3*a*B)*x^(7/2))/(a
*b^2*Sqrt[a + b*x]) - (35*a*(2*A*b - 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/(8*b^5) + (35
*(2*A*b - 3*a*B)*x^(3/2)*Sqrt[a + b*x])/(12*b^4) - (7*(2*A*b - 3*a*B)*x^(5/2)*Sq
rt[a + b*x])/(3*a*b^3) + (35*a^2*(2*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[
a + b*x]])/(8*b^(11/2))

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Rubi [A]  time = 0.234094, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{35 a^2 (2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{11/2}}-\frac{35 a \sqrt{x} \sqrt{a+b x} (2 A b-3 a B)}{8 b^5}+\frac{35 x^{3/2} \sqrt{a+b x} (2 A b-3 a B)}{12 b^4}-\frac{7 x^{5/2} \sqrt{a+b x} (2 A b-3 a B)}{3 a b^3}+\frac{2 x^{7/2} (2 A b-3 a B)}{a b^2 \sqrt{a+b x}}+\frac{2 x^{9/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*x^(9/2))/(3*a*b*(a + b*x)^(3/2)) + (2*(2*A*b - 3*a*B)*x^(7/2))/(a
*b^2*Sqrt[a + b*x]) - (35*a*(2*A*b - 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/(8*b^5) + (35
*(2*A*b - 3*a*B)*x^(3/2)*Sqrt[a + b*x])/(12*b^4) - (7*(2*A*b - 3*a*B)*x^(5/2)*Sq
rt[a + b*x])/(3*a*b^3) + (35*a^2*(2*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[
a + b*x]])/(8*b^(11/2))

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Rubi in Sympy [A]  time = 24.7567, size = 194, normalized size = 0.97 \[ \frac{35 a^{2} \left (A b - \frac{3 B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{4 b^{\frac{11}{2}}} - \frac{35 a \sqrt{x} \sqrt{a + b x} \left (2 A b - 3 B a\right )}{8 b^{5}} + \frac{35 x^{\frac{3}{2}} \sqrt{a + b x} \left (A b - \frac{3 B a}{2}\right )}{6 b^{4}} + \frac{2 x^{\frac{9}{2}} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{\frac{3}{2}}} + \frac{4 x^{\frac{7}{2}} \left (A b - \frac{3 B a}{2}\right )}{a b^{2} \sqrt{a + b x}} - \frac{14 x^{\frac{5}{2}} \sqrt{a + b x} \left (A b - \frac{3 B a}{2}\right )}{3 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35*a**2*(A*b - 3*B*a/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/(4*b**(11/2)) - 35*
a*sqrt(x)*sqrt(a + b*x)*(2*A*b - 3*B*a)/(8*b**5) + 35*x**(3/2)*sqrt(a + b*x)*(A*
b - 3*B*a/2)/(6*b**4) + 2*x**(9/2)*(A*b - B*a)/(3*a*b*(a + b*x)**(3/2)) + 4*x**(
7/2)*(A*b - 3*B*a/2)/(a*b**2*sqrt(a + b*x)) - 14*x**(5/2)*sqrt(a + b*x)*(A*b - 3
*B*a/2)/(3*a*b**3)

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Mathematica [A]  time = 0.188745, size = 136, normalized size = 0.68 \[ \frac{\sqrt{x} \left (315 a^4 B-210 a^3 b (A-2 B x)+7 a^2 b^2 x (9 B x-40 A)-6 a b^3 x^2 (7 A+3 B x)+4 b^4 x^3 (3 A+2 B x)\right )}{24 b^5 (a+b x)^{3/2}}-\frac{35 a^2 (3 a B-2 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{8 b^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(315*a^4*B - 210*a^3*b*(A - 2*B*x) + 4*b^4*x^3*(3*A + 2*B*x) - 6*a*b^3*
x^2*(7*A + 3*B*x) + 7*a^2*b^2*x*(-40*A + 9*B*x)))/(24*b^5*(a + b*x)^(3/2)) - (35
*a^2*(-2*A*b + 3*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(8*b^(11/2))

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Maple [B]  time = 0.031, size = 406, normalized size = 2. \[{\frac{1}{48} \left ( 16\,B{x}^{4}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }+24\,A{x}^{3}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-36\,B{x}^{3}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+210\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}{a}^{2}{b}^{3}-84\,A{x}^{2}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-315\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}{a}^{3}{b}^{2}+126\,B{x}^{2}{a}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+420\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{3}{b}^{2}-560\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}-630\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{4}b+840\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+210\,A{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-420\,A{a}^{3}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-315\,B{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +630\,B{a}^{4}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ) \sqrt{x}{b}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(16*B*x^4*b^(9/2)*(x*(b*x+a))^(1/2)+24*A*x^3*b^(9/2)*(x*(b*x+a))^(1/2)-36*B
*x^3*a*b^(7/2)*(x*(b*x+a))^(1/2)+210*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x
+a)/b^(1/2))*x^2*a^2*b^3-84*A*x^2*a*b^(7/2)*(x*(b*x+a))^(1/2)-315*B*ln(1/2*(2*(x
*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x^2*a^3*b^2+126*B*x^2*a^2*b^(5/2)*(x*(
b*x+a))^(1/2)+420*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x*a^3*
b^2-560*A*a^2*(x*(b*x+a))^(1/2)*x*b^(5/2)-630*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1
/2)+2*b*x+a)/b^(1/2))*x*a^4*b+840*B*a^3*(x*(b*x+a))^(1/2)*x*b^(3/2)+210*A*a^4*ln
(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-420*A*a^3*(x*(b*x+a))^(1/2
)*b^(3/2)-315*B*a^5*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+630*B*
a^4*(x*(b*x+a))^(1/2)*b^(1/2))/b^(11/2)*x^(1/2)/(x*(b*x+a))^(1/2)/(b*x+a)^(3/2)

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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)*x^(7/2)/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249162, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (3 \, B a^{4} - 2 \, A a^{3} b +{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{x} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (8 \, B b^{4} x^{5} - 6 \,{\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 21 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 140 \,{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 105 \,{\left (3 \, B a^{4} - 2 \, A a^{3} b\right )} x\right )} \sqrt{b}}{48 \,{\left (b^{6} x + a b^{5}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}, -\frac{105 \,{\left (3 \, B a^{4} - 2 \, A a^{3} b +{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (8 \, B b^{4} x^{5} - 6 \,{\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 21 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 140 \,{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 105 \,{\left (3 \, B a^{4} - 2 \, A a^{3} b\right )} x\right )} \sqrt{-b}}{24 \,{\left (b^{6} x + a b^{5}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(105*(3*B*a^4 - 2*A*a^3*b + (3*B*a^3*b - 2*A*a^2*b^2)*x)*sqrt(b*x + a)*sq
rt(x)*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)) - 2*(8*B*b^4*x^5 - 6*
(3*B*a*b^3 - 2*A*b^4)*x^4 + 21*(3*B*a^2*b^2 - 2*A*a*b^3)*x^3 + 140*(3*B*a^3*b -
2*A*a^2*b^2)*x^2 + 105*(3*B*a^4 - 2*A*a^3*b)*x)*sqrt(b))/((b^6*x + a*b^5)*sqrt(b
*x + a)*sqrt(b)*sqrt(x)), -1/24*(105*(3*B*a^4 - 2*A*a^3*b + (3*B*a^3*b - 2*A*a^2
*b^2)*x)*sqrt(b*x + a)*sqrt(x)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (8*B
*b^4*x^5 - 6*(3*B*a*b^3 - 2*A*b^4)*x^4 + 21*(3*B*a^2*b^2 - 2*A*a*b^3)*x^3 + 140*
(3*B*a^3*b - 2*A*a^2*b^2)*x^2 + 105*(3*B*a^4 - 2*A*a^3*b)*x)*sqrt(-b))/((b^6*x +
a*b^5)*sqrt(b*x + a)*sqrt(-b)*sqrt(x))]

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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**(7/2)*(B*x+A)/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.306289, size = 516, normalized size = 2.58 \[ \frac{1}{24} \, \sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )} B{\left | b \right |}}{b^{7}} - \frac{25 \, B a b^{20}{\left | b \right |} - 6 \, A b^{21}{\left | b \right |}}{b^{27}}\right )} + \frac{3 \,{\left (55 \, B a^{2} b^{20}{\left | b \right |} - 26 \, A a b^{21}{\left | b \right |}\right )}}{b^{27}}\right )} + \frac{35 \,{\left (3 \, B a^{3} \sqrt{b}{\left | b \right |} - 2 \, A a^{2} b^{\frac{3}{2}}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{16 \, b^{7}} + \frac{4 \,{\left (15 \, B a^{4}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b}{\left | b \right |} + 24 \, B a^{5}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}}{\left | b \right |} - 12 \, A a^{3}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{3}{2}}{\left | b \right |} + 13 \, B a^{6} b^{\frac{5}{2}}{\left | b \right |} - 18 \, A a^{4}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{5}{2}}{\left | b \right |} - 10 \, A a^{5} b^{\frac{7}{2}}{\left | b \right |}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*B*abs(b)/b^
7 - (25*B*a*b^20*abs(b) - 6*A*b^21*abs(b))/b^27) + 3*(55*B*a^2*b^20*abs(b) - 26*
A*a*b^21*abs(b))/b^27) + 35/16*(3*B*a^3*sqrt(b)*abs(b) - 2*A*a^2*b^(3/2)*abs(b))
*ln((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^7 + 4/3*(15*B*a^4*(sq
rt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*sqrt(b)*abs(b) + 24*B*a^5*(sqrt
(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(3/2)*abs(b) - 12*A*a^3*(sqrt(b
*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(3/2)*abs(b) + 13*B*a^6*b^(5/2)*a
bs(b) - 18*A*a^4*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(5/2)*abs
(b) - 10*A*a^5*b^(7/2)*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b
))^2 + a*b)^3*b^6)