3.514 \(\int \frac{x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=200 \[ -\frac{35 a^3 (8 A b-9 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{11/2}}+\frac{35 a^2 \sqrt{x} \sqrt{a+b x} (8 A b-9 a B)}{64 b^5}-\frac{35 a x^{3/2} \sqrt{a+b x} (8 A b-9 a B)}{96 b^4}+\frac{7 x^{5/2} \sqrt{a+b x} (8 A b-9 a B)}{24 b^3}-\frac{x^{7/2} \sqrt{a+b x} (8 A b-9 a B)}{4 a b^2}+\frac{2 x^{9/2} (A b-a B)}{a b \sqrt{a+b x}} \]

[Out]

(2*(A*b - a*B)*x^(9/2))/(a*b*Sqrt[a + b*x]) + (35*a^2*(8*A*b - 9*a*B)*Sqrt[x]*Sq
rt[a + b*x])/(64*b^5) - (35*a*(8*A*b - 9*a*B)*x^(3/2)*Sqrt[a + b*x])/(96*b^4) +
(7*(8*A*b - 9*a*B)*x^(5/2)*Sqrt[a + b*x])/(24*b^3) - ((8*A*b - 9*a*B)*x^(7/2)*Sq
rt[a + b*x])/(4*a*b^2) - (35*a^3*(8*A*b - 9*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[
a + b*x]])/(64*b^(11/2))

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Rubi [A]  time = 0.23822, antiderivative size = 200, 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 \[ -\frac{35 a^3 (8 A b-9 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{11/2}}+\frac{35 a^2 \sqrt{x} \sqrt{a+b x} (8 A b-9 a B)}{64 b^5}-\frac{35 a x^{3/2} \sqrt{a+b x} (8 A b-9 a B)}{96 b^4}+\frac{7 x^{5/2} \sqrt{a+b x} (8 A b-9 a B)}{24 b^3}-\frac{x^{7/2} \sqrt{a+b x} (8 A b-9 a B)}{4 a b^2}+\frac{2 x^{9/2} (A b-a B)}{a b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(x^(7/2)*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(2*(A*b - a*B)*x^(9/2))/(a*b*Sqrt[a + b*x]) + (35*a^2*(8*A*b - 9*a*B)*Sqrt[x]*Sq
rt[a + b*x])/(64*b^5) - (35*a*(8*A*b - 9*a*B)*x^(3/2)*Sqrt[a + b*x])/(96*b^4) +
(7*(8*A*b - 9*a*B)*x^(5/2)*Sqrt[a + b*x])/(24*b^3) - ((8*A*b - 9*a*B)*x^(7/2)*Sq
rt[a + b*x])/(4*a*b^2) - (35*a^3*(8*A*b - 9*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[
a + b*x]])/(64*b^(11/2))

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Rubi in Sympy [A]  time = 24.7491, size = 194, normalized size = 0.97 \[ - \frac{35 a^{3} \left (8 A b - 9 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{64 b^{\frac{11}{2}}} + \frac{35 a^{2} \sqrt{x} \sqrt{a + b x} \left (8 A b - 9 B a\right )}{64 b^{5}} - \frac{35 a x^{\frac{3}{2}} \sqrt{a + b x} \left (8 A b - 9 B a\right )}{96 b^{4}} + \frac{7 x^{\frac{5}{2}} \sqrt{a + b x} \left (8 A b - 9 B a\right )}{24 b^{3}} + \frac{2 x^{\frac{9}{2}} \left (A b - B a\right )}{a b \sqrt{a + b x}} - \frac{x^{\frac{7}{2}} \sqrt{a + b x} \left (8 A b - 9 B a\right )}{4 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x+A)/(b*x+a)**(3/2),x)

[Out]

-35*a**3*(8*A*b - 9*B*a)*atanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/(64*b**(11/2)) + 3
5*a**2*sqrt(x)*sqrt(a + b*x)*(8*A*b - 9*B*a)/(64*b**5) - 35*a*x**(3/2)*sqrt(a +
b*x)*(8*A*b - 9*B*a)/(96*b**4) + 7*x**(5/2)*sqrt(a + b*x)*(8*A*b - 9*B*a)/(24*b*
*3) + 2*x**(9/2)*(A*b - B*a)/(a*b*sqrt(a + b*x)) - x**(7/2)*sqrt(a + b*x)*(8*A*b
- 9*B*a)/(4*a*b**2)

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Mathematica [A]  time = 0.235244, size = 138, normalized size = 0.69 \[ \frac{35 a^3 (9 a B-8 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{64 b^{11/2}}+\frac{\sqrt{x} \left (-945 a^4 B+105 a^3 b (8 A-3 B x)+14 a^2 b^2 x (20 A+9 B x)-8 a b^3 x^2 (14 A+9 B x)+16 b^4 x^3 (4 A+3 B x)\right )}{192 b^5 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(7/2)*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(Sqrt[x]*(-945*a^4*B + 105*a^3*b*(8*A - 3*B*x) + 16*b^4*x^3*(4*A + 3*B*x) - 8*a*
b^3*x^2*(14*A + 9*B*x) + 14*a^2*b^2*x*(20*A + 9*B*x)))/(192*b^5*Sqrt[a + b*x]) +
(35*a^3*(-8*A*b + 9*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(64*b^(11/2))

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Maple [B]  time = 0.033, size = 330, normalized size = 1.7 \[ -{\frac{1}{384} \left ( -96\,B{x}^{4}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-128\,A{x}^{3}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }+144\,B{x}^{3}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+224\,A{x}^{2}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-252\,B{x}^{2}{a}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+840\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{3}{b}^{2}-560\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}-945\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{4}b+630\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+840\,A{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-1680\,A{a}^{3}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-945\,B{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +1890\,B{a}^{4}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ) \sqrt{x}{b}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x+A)/(b*x+a)^(3/2),x)

[Out]

-1/384*(-96*B*x^4*b^(9/2)*(x*(b*x+a))^(1/2)-128*A*x^3*b^(9/2)*(x*(b*x+a))^(1/2)+
144*B*x^3*a*b^(7/2)*(x*(b*x+a))^(1/2)+224*A*x^2*a*b^(7/2)*(x*(b*x+a))^(1/2)-252*
B*x^2*a^2*b^(5/2)*(x*(b*x+a))^(1/2)+840*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)-945*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+630*B*a^3*(x*(b*x+a))^(1/2)*
x*b^(3/2)+840*A*a^4*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-1680
*A*a^3*(x*(b*x+a))^(1/2)*b^(3/2)-945*B*a^5*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2
*b*x+a)/b^(1/2))+1890*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)^(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)*x^(7/2)/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252899, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (9 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt{b x + a} \sqrt{x} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (48 \, B b^{4} x^{5} - 8 \,{\left (9 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} + 14 \,{\left (9 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 35 \,{\left (9 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} - 105 \,{\left (9 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )} \sqrt{b}}{384 \, \sqrt{b x + a} b^{\frac{11}{2}} \sqrt{x}}, \frac{105 \,{\left (9 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt{b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (48 \, B b^{4} x^{5} - 8 \,{\left (9 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} + 14 \,{\left (9 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 35 \,{\left (9 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} - 105 \,{\left (9 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )} \sqrt{-b}}{192 \, \sqrt{b x + a} \sqrt{-b} b^{5} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(7/2)/(b*x + a)^(3/2),x, algorithm="fricas")

[Out]

[-1/384*(105*(9*B*a^4 - 8*A*a^3*b)*sqrt(b*x + a)*sqrt(x)*log(-2*sqrt(b*x + a)*b*
sqrt(x) + (2*b*x + a)*sqrt(b)) - 2*(48*B*b^4*x^5 - 8*(9*B*a*b^3 - 8*A*b^4)*x^4 +
14*(9*B*a^2*b^2 - 8*A*a*b^3)*x^3 - 35*(9*B*a^3*b - 8*A*a^2*b^2)*x^2 - 105*(9*B*
a^4 - 8*A*a^3*b)*x)*sqrt(b))/(sqrt(b*x + a)*b^(11/2)*sqrt(x)), 1/192*(105*(9*B*a
^4 - 8*A*a^3*b)*sqrt(b*x + a)*sqrt(x)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x)))
+ (48*B*b^4*x^5 - 8*(9*B*a*b^3 - 8*A*b^4)*x^4 + 14*(9*B*a^2*b^2 - 8*A*a*b^3)*x^
3 - 35*(9*B*a^3*b - 8*A*a^2*b^2)*x^2 - 105*(9*B*a^4 - 8*A*a^3*b)*x)*sqrt(-b))/(s
qrt(b*x + a)*sqrt(-b)*b^5*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27467, size = 340, normalized size = 1.7 \[ \frac{1}{192} \,{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )} B{\left | b \right |}}{b^{7}} - \frac{33 \, B a b^{27}{\left | b \right |} - 8 \, A b^{28}{\left | b \right |}}{b^{34}}\right )} + \frac{315 \, B a^{2} b^{27}{\left | b \right |} - 152 \, A a b^{28}{\left | b \right |}}{b^{34}}\right )} - \frac{3 \,{\left (325 \, B a^{3} b^{27}{\left | b \right |} - 232 \, A a^{2} b^{28}{\left | b \right |}\right )}}{b^{34}}\right )} \sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a} - \frac{35 \,{\left (9 \, B a^{4} \sqrt{b}{\left | b \right |} - 8 \, A a^{3} b^{\frac{3}{2}}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{128 \, b^{7}} - \frac{4 \,{\left (B a^{5} \sqrt{b}{\left | b \right |} - A a^{4} b^{\frac{3}{2}}{\left | b \right |}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(7/2)/(b*x + a)^(3/2),x, algorithm="giac")

[Out]

1/192*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)*B*abs(b)/b^7 - (33*B*a*b^27*abs(b)
- 8*A*b^28*abs(b))/b^34) + (315*B*a^2*b^27*abs(b) - 152*A*a*b^28*abs(b))/b^34) -
3*(325*B*a^3*b^27*abs(b) - 232*A*a^2*b^28*abs(b))/b^34)*sqrt((b*x + a)*b - a*b)
*sqrt(b*x + a) - 35/128*(9*B*a^4*sqrt(b)*abs(b) - 8*A*a^3*b^(3/2)*abs(b))*ln((sq
rt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^7 - 4*(B*a^5*sqrt(b)*abs(b)
- A*a^4*b^(3/2)*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 +
a*b)*b^6)