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

Optimal. Leaf size=316 \[ \frac{20 a^2 b^2 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{2 b^4 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{5 (a+b x)}+\frac{10 a b^3 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{3 (a+b x)}+\frac{2 b^5 B x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}-\frac{2 a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac{2 a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 x^{3/2} (a+b x)}-\frac{10 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{\sqrt{x} (a+b x)} \]

[Out]

(-2*a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^(5/2)*(a + b*x)) - (2*a^4*(5*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^(3/2)*(a + b*x)) - (10*a^3*b*(2*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(Sqrt[x]*(a + b*x)) + (20*a^2*b^2*(A*b + a*B
)*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (10*a*b^3*(A*b + 2*a*B)*x^(
3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*b^4*(A*b + 5*a*B)*x^(5/2)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (2*b^5*B*x^(7/2)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(7*(a + b*x))

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Rubi [A]  time = 0.337526, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{20 a^2 b^2 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{2 b^4 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{5 (a+b x)}+\frac{10 a b^3 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{3 (a+b x)}+\frac{2 b^5 B x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}-\frac{2 a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac{2 a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 x^{3/2} (a+b x)}-\frac{10 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{\sqrt{x} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^(5/2)*(a + b*x)) - (2*a^4*(5*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^(3/2)*(a + b*x)) - (10*a^3*b*(2*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(Sqrt[x]*(a + b*x)) + (20*a^2*b^2*(A*b + a*B
)*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (10*a*b^3*(A*b + 2*a*B)*x^(
3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*b^4*(A*b + 5*a*B)*x^(5/2)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (2*b^5*B*x^(7/2)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(7*(a + b*x))

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Rubi in Sympy [A]  time = 36.1034, size = 318, normalized size = 1.01 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 a x^{\frac{5}{2}}} + \frac{512 a^{2} b^{2} \sqrt{x} \left (7 A b + 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{105 \left (a + b x\right )} + \frac{256 a b^{2} \sqrt{x} \left (7 A b + 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{105} + \frac{64 b^{2} \sqrt{x} \left (3 a + 3 b x\right ) \left (7 A b + 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{105} + \frac{32 b^{2} \sqrt{x} \left (7 A b + 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{21 a} - \frac{4 b \left (a + b x\right ) \left (7 A b + 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 a \sqrt{x}} - \frac{2 \left (7 A b + 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{15 a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(5*a*x**(5/2)) + 512*a**2*b
**2*sqrt(x)*(7*A*b + 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(105*(a + b*x)) + 2
56*a*b**2*sqrt(x)*(7*A*b + 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/105 + 64*b**2
*sqrt(x)*(3*a + 3*b*x)*(7*A*b + 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/105 + 32
*b**2*sqrt(x)*(7*A*b + 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(21*a) - 4*b*(
a + b*x)*(7*A*b + 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(3*a*sqrt(x)) - 2*(
7*A*b + 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(15*a*x**(3/2))

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Mathematica [A]  time = 0.0937477, size = 122, normalized size = 0.39 \[ -\frac{2 \sqrt{(a+b x)^2} \left (7 a^5 (3 A+5 B x)+175 a^4 b x (A+3 B x)+1050 a^3 b^2 x^2 (A-B x)-350 a^2 b^3 x^3 (3 A+B x)-35 a b^4 x^4 (5 A+3 B x)-3 b^5 x^5 (7 A+5 B x)\right )}{105 x^{5/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[(a + b*x)^2]*(1050*a^3*b^2*x^2*(A - B*x) - 350*a^2*b^3*x^3*(3*A + B*x)
+ 175*a^4*b*x*(A + 3*B*x) - 35*a*b^4*x^4*(5*A + 3*B*x) + 7*a^5*(3*A + 5*B*x) - 3
*b^5*x^5*(7*A + 5*B*x)))/(105*x^(5/2)*(a + b*x))

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Maple [A]  time = 0.012, size = 140, normalized size = 0.4 \[ -{\frac{-30\,B{b}^{5}{x}^{6}-42\,A{x}^{5}{b}^{5}-210\,B{x}^{5}a{b}^{4}-350\,A{x}^{4}a{b}^{4}-700\,B{x}^{4}{a}^{2}{b}^{3}-2100\,A{x}^{3}{a}^{2}{b}^{3}-2100\,B{x}^{3}{a}^{3}{b}^{2}+2100\,A{x}^{2}{a}^{3}{b}^{2}+1050\,B{x}^{2}{a}^{4}b+350\,Ax{a}^{4}b+70\,Bx{a}^{5}+42\,A{a}^{5}}{105\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(-15*B*b^5*x^6-21*A*b^5*x^5-105*B*a*b^4*x^5-175*A*a*b^4*x^4-350*B*a^2*b^3
*x^4-1050*A*a^2*b^3*x^3-1050*B*a^3*b^2*x^3+1050*A*a^3*b^2*x^2+525*B*a^4*b*x^2+17
5*A*a^4*b*x+35*B*a^5*x+21*A*a^5)*((b*x+a)^2)^(5/2)/x^(5/2)/(b*x+a)^5

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Maxima [A]  time = 0.712104, size = 316, normalized size = 1. \[ \frac{2}{15} \,{\left ({\left (3 \, b^{5} x^{2} + 5 \, a b^{4} x\right )} \sqrt{x} + \frac{20 \,{\left (a b^{4} x^{2} + 3 \, a^{2} b^{3} x\right )}}{\sqrt{x}} + \frac{90 \,{\left (a^{2} b^{3} x^{2} - a^{3} b^{2} x\right )}}{x^{\frac{3}{2}}} - \frac{20 \,{\left (3 \, a^{3} b^{2} x^{2} + a^{4} b x\right )}}{x^{\frac{5}{2}}} - \frac{5 \, a^{4} b x^{2} + 3 \, a^{5} x}{x^{\frac{7}{2}}}\right )} A + \frac{2}{105} \,{\left (3 \,{\left (5 \, b^{5} x^{2} + 7 \, a b^{4} x\right )} x^{\frac{3}{2}} + 28 \,{\left (3 \, a b^{4} x^{2} + 5 \, a^{2} b^{3} x\right )} \sqrt{x} + \frac{210 \,{\left (a^{2} b^{3} x^{2} + 3 \, a^{3} b^{2} x\right )}}{\sqrt{x}} + \frac{420 \,{\left (a^{3} b^{2} x^{2} - a^{4} b x\right )}}{x^{\frac{3}{2}}} - \frac{35 \,{\left (3 \, a^{4} b x^{2} + a^{5} x\right )}}{x^{\frac{5}{2}}}\right )} B \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*((3*b^5*x^2 + 5*a*b^4*x)*sqrt(x) + 20*(a*b^4*x^2 + 3*a^2*b^3*x)/sqrt(x) + 9
0*(a^2*b^3*x^2 - a^3*b^2*x)/x^(3/2) - 20*(3*a^3*b^2*x^2 + a^4*b*x)/x^(5/2) - (5*
a^4*b*x^2 + 3*a^5*x)/x^(7/2))*A + 2/105*(3*(5*b^5*x^2 + 7*a*b^4*x)*x^(3/2) + 28*
(3*a*b^4*x^2 + 5*a^2*b^3*x)*sqrt(x) + 210*(a^2*b^3*x^2 + 3*a^3*b^2*x)/sqrt(x) +
420*(a^3*b^2*x^2 - a^4*b*x)/x^(3/2) - 35*(3*a^4*b*x^2 + a^5*x)/x^(5/2))*B

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Fricas [A]  time = 0.285342, size = 161, normalized size = 0.51 \[ \frac{2 \,{\left (15 \, B b^{5} x^{6} - 21 \, A a^{5} + 21 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 175 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 1050 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 525 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 35 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x\right )}}{105 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*B*b^5*x^6 - 21*A*a^5 + 21*(5*B*a*b^4 + A*b^5)*x^5 + 175*(2*B*a^2*b^3 +
A*a*b^4)*x^4 + 1050*(B*a^3*b^2 + A*a^2*b^3)*x^3 - 525*(B*a^4*b + 2*A*a^3*b^2)*x
^2 - 35*(B*a^5 + 5*A*a^4*b)*x)/x^(5/2)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276335, size = 265, normalized size = 0.84 \[ \frac{2}{7} \, B b^{5} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + 2 \, B a b^{4} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{5} \, A b^{5} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{3} \, B a^{2} b^{3} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{3} \, A a b^{4} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 20 \, B a^{3} b^{2} \sqrt{x}{\rm sign}\left (b x + a\right ) + 20 \, A a^{2} b^{3} \sqrt{x}{\rm sign}\left (b x + a\right ) - \frac{2 \,{\left (75 \, B a^{4} b x^{2}{\rm sign}\left (b x + a\right ) + 150 \, A a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 5 \, B a^{5} x{\rm sign}\left (b x + a\right ) + 25 \, A a^{4} b x{\rm sign}\left (b x + a\right ) + 3 \, A a^{5}{\rm sign}\left (b x + a\right )\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*B*b^5*x^(7/2)*sign(b*x + a) + 2*B*a*b^4*x^(5/2)*sign(b*x + a) + 2/5*A*b^5*x^
(5/2)*sign(b*x + a) + 20/3*B*a^2*b^3*x^(3/2)*sign(b*x + a) + 10/3*A*a*b^4*x^(3/2
)*sign(b*x + a) + 20*B*a^3*b^2*sqrt(x)*sign(b*x + a) + 20*A*a^2*b^3*sqrt(x)*sign
(b*x + a) - 2/15*(75*B*a^4*b*x^2*sign(b*x + a) + 150*A*a^3*b^2*x^2*sign(b*x + a)
+ 5*B*a^5*x*sign(b*x + a) + 25*A*a^4*b*x*sign(b*x + a) + 3*A*a^5*sign(b*x + a))
/x^(5/2)