3.225 \(\int \frac{x^{7/2} (A+B x)}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=170 \[ \frac{32 b^3 \sqrt{b x+c x^2} (8 b B-9 A c)}{315 c^5 \sqrt{x}}-\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2} (8 b B-9 A c)}{315 c^4}+\frac{4 b x^{3/2} \sqrt{b x+c x^2} (8 b B-9 A c)}{105 c^3}-\frac{2 x^{5/2} \sqrt{b x+c x^2} (8 b B-9 A c)}{63 c^2}+\frac{2 B x^{7/2} \sqrt{b x+c x^2}}{9 c} \]

[Out]

(32*b^3*(8*b*B - 9*A*c)*Sqrt[b*x + c*x^2])/(315*c^5*Sqrt[x]) - (16*b^2*(8*b*B -
9*A*c)*Sqrt[x]*Sqrt[b*x + c*x^2])/(315*c^4) + (4*b*(8*b*B - 9*A*c)*x^(3/2)*Sqrt[
b*x + c*x^2])/(105*c^3) - (2*(8*b*B - 9*A*c)*x^(5/2)*Sqrt[b*x + c*x^2])/(63*c^2)
+ (2*B*x^(7/2)*Sqrt[b*x + c*x^2])/(9*c)

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Rubi [A]  time = 0.34336, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{32 b^3 \sqrt{b x+c x^2} (8 b B-9 A c)}{315 c^5 \sqrt{x}}-\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2} (8 b B-9 A c)}{315 c^4}+\frac{4 b x^{3/2} \sqrt{b x+c x^2} (8 b B-9 A c)}{105 c^3}-\frac{2 x^{5/2} \sqrt{b x+c x^2} (8 b B-9 A c)}{63 c^2}+\frac{2 B x^{7/2} \sqrt{b x+c x^2}}{9 c} \]

Antiderivative was successfully verified.

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

[Out]

(32*b^3*(8*b*B - 9*A*c)*Sqrt[b*x + c*x^2])/(315*c^5*Sqrt[x]) - (16*b^2*(8*b*B -
9*A*c)*Sqrt[x]*Sqrt[b*x + c*x^2])/(315*c^4) + (4*b*(8*b*B - 9*A*c)*x^(3/2)*Sqrt[
b*x + c*x^2])/(105*c^3) - (2*(8*b*B - 9*A*c)*x^(5/2)*Sqrt[b*x + c*x^2])/(63*c^2)
+ (2*B*x^(7/2)*Sqrt[b*x + c*x^2])/(9*c)

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Rubi in Sympy [A]  time = 21.5597, size = 167, normalized size = 0.98 \[ \frac{2 B x^{\frac{7}{2}} \sqrt{b x + c x^{2}}}{9 c} - \frac{32 b^{3} \left (9 A c - 8 B b\right ) \sqrt{b x + c x^{2}}}{315 c^{5} \sqrt{x}} + \frac{16 b^{2} \sqrt{x} \left (9 A c - 8 B b\right ) \sqrt{b x + c x^{2}}}{315 c^{4}} - \frac{4 b x^{\frac{3}{2}} \left (9 A c - 8 B b\right ) \sqrt{b x + c x^{2}}}{105 c^{3}} + \frac{2 x^{\frac{5}{2}} \left (9 A c - 8 B b\right ) \sqrt{b x + c x^{2}}}{63 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x+A)/(c*x**2+b*x)**(1/2),x)

[Out]

2*B*x**(7/2)*sqrt(b*x + c*x**2)/(9*c) - 32*b**3*(9*A*c - 8*B*b)*sqrt(b*x + c*x**
2)/(315*c**5*sqrt(x)) + 16*b**2*sqrt(x)*(9*A*c - 8*B*b)*sqrt(b*x + c*x**2)/(315*
c**4) - 4*b*x**(3/2)*(9*A*c - 8*B*b)*sqrt(b*x + c*x**2)/(105*c**3) + 2*x**(5/2)*
(9*A*c - 8*B*b)*sqrt(b*x + c*x**2)/(63*c**2)

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Mathematica [A]  time = 0.100039, size = 94, normalized size = 0.55 \[ \frac{2 \sqrt{x (b+c x)} \left (-16 b^3 c (9 A+4 B x)+24 b^2 c^2 x (3 A+2 B x)-2 b c^3 x^2 (27 A+20 B x)+5 c^4 x^3 (9 A+7 B x)+128 b^4 B\right )}{315 c^5 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x*(b + c*x)]*(128*b^4*B + 24*b^2*c^2*x*(3*A + 2*B*x) - 16*b^3*c*(9*A + 4
*B*x) + 5*c^4*x^3*(9*A + 7*B*x) - 2*b*c^3*x^2*(27*A + 20*B*x)))/(315*c^5*Sqrt[x]
)

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Maple [A]  time = 0.009, size = 107, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -35\,B{x}^{4}{c}^{4}-45\,A{c}^{4}{x}^{3}+40\,Bb{c}^{3}{x}^{3}+54\,Ab{c}^{3}{x}^{2}-48\,B{b}^{2}{c}^{2}{x}^{2}-72\,A{b}^{2}{c}^{2}x+64\,B{b}^{3}cx+144\,A{b}^{3}c-128\,{b}^{4}B \right ) }{315\,{c}^{5}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x+A)/(c*x^2+b*x)^(1/2),x)

[Out]

-2/315*(c*x+b)*(-35*B*c^4*x^4-45*A*c^4*x^3+40*B*b*c^3*x^3+54*A*b*c^3*x^2-48*B*b^
2*c^2*x^2-72*A*b^2*c^2*x+64*B*b^3*c*x+144*A*b^3*c-128*B*b^4)*x^(1/2)/c^5/(c*x^2+
b*x)^(1/2)

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Maxima [A]  time = 0.701113, size = 162, normalized size = 0.95 \[ \frac{2 \,{\left (5 \, c^{4} x^{4} - b c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} - 8 \, b^{3} c x - 16 \, b^{4}\right )} A}{35 \, \sqrt{c x + b} c^{4}} + \frac{2 \,{\left (35 \, c^{5} x^{5} - 5 \, b c^{4} x^{4} + 8 \, b^{2} c^{3} x^{3} - 16 \, b^{3} c^{2} x^{2} + 64 \, b^{4} c x + 128 \, b^{5}\right )} B}{315 \, \sqrt{c x + b} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(7/2)/sqrt(c*x^2 + b*x),x, algorithm="maxima")

[Out]

2/35*(5*c^4*x^4 - b*c^3*x^3 + 2*b^2*c^2*x^2 - 8*b^3*c*x - 16*b^4)*A/(sqrt(c*x +
b)*c^4) + 2/315*(35*c^5*x^5 - 5*b*c^4*x^4 + 8*b^2*c^3*x^3 - 16*b^3*c^2*x^2 + 64*
b^4*c*x + 128*b^5)*B/(sqrt(c*x + b)*c^5)

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Fricas [A]  time = 0.29185, size = 177, normalized size = 1.04 \[ \frac{2 \,{\left (35 \, B c^{5} x^{6} - 5 \,{\left (B b c^{4} - 9 \, A c^{5}\right )} x^{5} +{\left (8 \, B b^{2} c^{3} - 9 \, A b c^{4}\right )} x^{4} - 2 \,{\left (8 \, B b^{3} c^{2} - 9 \, A b^{2} c^{3}\right )} x^{3} + 8 \,{\left (8 \, B b^{4} c - 9 \, A b^{3} c^{2}\right )} x^{2} + 16 \,{\left (8 \, B b^{5} - 9 \, A b^{4} c\right )} x\right )}}{315 \, \sqrt{c x^{2} + b x} c^{5} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*c^5*x^6 - 5*(B*b*c^4 - 9*A*c^5)*x^5 + (8*B*b^2*c^3 - 9*A*b*c^4)*x^4
- 2*(8*B*b^3*c^2 - 9*A*b^2*c^3)*x^3 + 8*(8*B*b^4*c - 9*A*b^3*c^2)*x^2 + 16*(8*B*
b^5 - 9*A*b^4*c)*x)/(sqrt(c*x^2 + b*x)*c^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)/(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279868, size = 185, normalized size = 1.09 \[ \frac{2 \,{\left (35 \,{\left (c x + b\right )}^{\frac{9}{2}} B - 180 \,{\left (c x + b\right )}^{\frac{7}{2}} B b + 378 \,{\left (c x + b\right )}^{\frac{5}{2}} B b^{2} - 420 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{3} + 315 \, \sqrt{c x + b} B b^{4} + 45 \,{\left (c x + b\right )}^{\frac{7}{2}} A c - 189 \,{\left (c x + b\right )}^{\frac{5}{2}} A b c + 315 \,{\left (c x + b\right )}^{\frac{3}{2}} A b^{2} c - 315 \, \sqrt{c x + b} A b^{3} c\right )}}{315 \, c^{5}} - \frac{32 \,{\left (8 \, B b^{\frac{9}{2}} - 9 \, A b^{\frac{7}{2}} c\right )}}{315 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*(c*x + b)^(9/2)*B - 180*(c*x + b)^(7/2)*B*b + 378*(c*x + b)^(5/2)*B*b^
2 - 420*(c*x + b)^(3/2)*B*b^3 + 315*sqrt(c*x + b)*B*b^4 + 45*(c*x + b)^(7/2)*A*c
- 189*(c*x + b)^(5/2)*A*b*c + 315*(c*x + b)^(3/2)*A*b^2*c - 315*sqrt(c*x + b)*A
*b^3*c)/c^5 - 32/315*(8*B*b^(9/2) - 9*A*b^(7/2)*c)/c^5