\(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^{9/2}} \, dx\) [808]

Optimal result

Integrand size = 31, antiderivative size = 316 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{9/2}} \, dx=-\frac {2 a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac {2 a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac {10 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac {20 a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {10 a b^3 (A b+2 a B) \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 b^4 (A b+5 a B) x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b^5 B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]

[Out]

-2/7*a^5*A*((b*x+a)^2)^(1/2)/x^(7/2)/(b*x+a)-2/5*a^4*(5*A*b+B*a)*((b*x+a)^2)^(1/2)/x^(5/2)/(b*x+a)-10/3*a^3*b*
(2*A*b+B*a)*((b*x+a)^2)^(1/2)/x^(3/2)/(b*x+a)+2/3*b^4*(A*b+5*B*a)*x^(3/2)*((b*x+a)^2)^(1/2)/(b*x+a)+2/5*b^5*B*
x^(5/2)*((b*x+a)^2)^(1/2)/(b*x+a)-20*a^2*b^2*(A*b+B*a)*((b*x+a)^2)^(1/2)/(b*x+a)/x^(1/2)+10*a*b^3*(A*b+2*B*a)*
x^(1/2)*((b*x+a)^2)^(1/2)/(b*x+a)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {784, 77} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{9/2}} \, dx=-\frac {20 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{\sqrt {x} (a+b x)}+\frac {2 b^4 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{3 (a+b x)}+\frac {10 a b^3 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac {2 b^5 B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}-\frac {2 a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac {2 a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{5 x^{5/2} (a+b x)}-\frac {10 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{3 x^{3/2} (a+b x)} \]

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^(9/2),x]

[Out]

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

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rule 784

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{x^{9/2}} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^5 A b^5}{x^{9/2}}+\frac {a^4 b^5 (5 A b+a B)}{x^{7/2}}+\frac {5 a^3 b^6 (2 A b+a B)}{x^{5/2}}+\frac {10 a^2 b^7 (A b+a B)}{x^{3/2}}+\frac {5 a b^8 (A b+2 a B)}{\sqrt {x}}+b^9 (A b+5 a B) \sqrt {x}+b^{10} B x^{3/2}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {2 a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac {2 a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac {10 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac {20 a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {10 a b^3 (A b+2 a B) \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 b^4 (A b+5 a B) x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b^5 B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.39 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{9/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (1050 a^2 b^3 x^3 (A-B x)-175 a b^4 x^4 (3 A+B x)+350 a^3 b^2 x^2 (A+3 B x)-7 b^5 x^5 (5 A+3 B x)+35 a^4 b x (3 A+5 B x)+3 a^5 (5 A+7 B x)\right )}{105 x^{7/2} (a+b x)} \]

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^(9/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.44

method	result	size
gosper	\(-\frac {2 \left (-21 B \,b^{5} x^{6}-35 A \,b^{5} x^{5}-175 B a \,b^{4} x^{5}-525 A a \,b^{4} x^{4}-1050 B \,a^{2} b^{3} x^{4}+1050 A \,a^{2} b^{3} x^{3}+1050 B \,a^{3} b^{2} x^{3}+350 A \,a^{3} b^{2} x^{2}+175 B \,a^{4} b \,x^{2}+105 A \,a^{4} b x +21 a^{5} B x +15 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{105 x^{\frac {7}{2}} \left (b x +a \right )^{5}}\)	\(140\)
default	\(-\frac {2 \left (-21 B \,b^{5} x^{6}-35 A \,b^{5} x^{5}-175 B a \,b^{4} x^{5}-525 A a \,b^{4} x^{4}-1050 B \,a^{2} b^{3} x^{4}+1050 A \,a^{2} b^{3} x^{3}+1050 B \,a^{3} b^{2} x^{3}+350 A \,a^{3} b^{2} x^{2}+175 B \,a^{4} b \,x^{2}+105 A \,a^{4} b x +21 a^{5} B x +15 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{105 x^{\frac {7}{2}} \left (b x +a \right )^{5}}\)	\(140\)
risch	\(-\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (-21 B \,b^{5} x^{6}-35 A \,b^{5} x^{5}-175 B a \,b^{4} x^{5}-525 A a \,b^{4} x^{4}-1050 B \,a^{2} b^{3} x^{4}+1050 A \,a^{2} b^{3} x^{3}+1050 B \,a^{3} b^{2} x^{3}+350 A \,a^{3} b^{2} x^{2}+175 B \,a^{4} b \,x^{2}+105 A \,a^{4} b x +21 a^{5} B x +15 A \,a^{5}\right )}{105 \left (b x +a \right ) x^{\frac {7}{2}}}\)	\(140\)

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/105*(-21*B*b^5*x^6-35*A*b^5*x^5-175*B*a*b^4*x^5-525*A*a*b^4*x^4-1050*B*a^2*b^3*x^4+1050*A*a^2*b^3*x^3+1050*
B*a^3*b^2*x^3+350*A*a^3*b^2*x^2+175*B*a^4*b*x^2+105*A*a^4*b*x+21*B*a^5*x+15*A*a^5)*((b*x+a)^2)^(5/2)/x^(7/2)/(
b*x+a)^5

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.38 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{9/2}} \, dx=\frac {2 \, {\left (21 \, B b^{5} x^{6} - 15 \, A a^{5} + 35 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 525 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 1050 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 175 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 21 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \]

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^(9/2),x, algorithm="fricas")

[Out]

2/105*(21*B*b^5*x^6 - 15*A*a^5 + 35*(5*B*a*b^4 + A*b^5)*x^5 + 525*(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 - 175*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 21*(B*a^5 + 5*A*a^4*b)*x)/x^(7/2)

Sympy [F]

\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{9/2}} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{\frac {9}{2}}}\, dx \]

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/x**(9/2),x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**(9/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.74 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{9/2}} \, dx=\frac {2}{15} \, {\left ({\left (3 \, b^{5} x^{2} + 5 \, a b^{4} x\right )} \sqrt {x} + \frac {20 \, {\left (a b^{4} x^{2} + 3 \, a^{2} b^{3} x\right )}}{\sqrt {x}} + \frac {90 \, {\left (a^{2} b^{3} x^{2} - a^{3} b^{2} x\right )}}{x^{\frac {3}{2}}} - \frac {20 \, {\left (3 \, a^{3} b^{2} x^{2} + a^{4} b x\right )}}{x^{\frac {5}{2}}} - \frac {5 \, a^{4} b x^{2} + 3 \, a^{5} x}{x^{\frac {7}{2}}}\right )} B + \frac {2}{105} \, A {\left (\frac {35 \, {\left (b^{5} x^{2} + 3 \, a b^{4} x\right )}}{\sqrt {x}} + \frac {420 \, {\left (a b^{4} x^{2} - a^{2} b^{3} x\right )}}{x^{\frac {3}{2}}} - \frac {210 \, {\left (3 \, a^{2} b^{3} x^{2} + a^{3} b^{2} x\right )}}{x^{\frac {5}{2}}} - \frac {28 \, {\left (5 \, a^{3} b^{2} x^{2} + 3 \, a^{4} b x\right )}}{x^{\frac {7}{2}}} - \frac {3 \, {\left (7 \, a^{4} b x^{2} + 5 \, a^{5} x\right )}}{x^{\frac {9}{2}}}\right )} \]

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^(9/2),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) + 90*(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))*B + 2/105*A*(35*(b^5*x^2 + 3*a
*b^4*x)/sqrt(x) + 420*(a*b^4*x^2 - a^2*b^3*x)/x^(3/2) - 210*(3*a^2*b^3*x^2 + a^3*b^2*x)/x^(5/2) - 28*(5*a^3*b^
2*x^2 + 3*a^4*b*x)/x^(7/2) - 3*(7*a^4*b*x^2 + 5*a^5*x)/x^(9/2))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.62 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{9/2}} \, dx=\frac {2}{5} \, B b^{5} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, B a b^{4} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, A b^{5} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 20 \, B a^{2} b^{3} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) + 10 \, A a b^{4} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) - \frac {2 \, {\left (1050 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 1050 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 175 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 350 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 105 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 15 \, A a^{5} \mathrm {sgn}\left (b x + a\right )\right )}}{105 \, x^{\frac {7}{2}}} \]

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^(9/2),x, algorithm="giac")

[Out]

2/5*B*b^5*x^(5/2)*sgn(b*x + a) + 10/3*B*a*b^4*x^(3/2)*sgn(b*x + a) + 2/3*A*b^5*x^(3/2)*sgn(b*x + a) + 20*B*a^2
*b^3*sqrt(x)*sgn(b*x + a) + 10*A*a*b^4*sqrt(x)*sgn(b*x + a) - 2/105*(1050*B*a^3*b^2*x^3*sgn(b*x + a) + 1050*A*
a^2*b^3*x^3*sgn(b*x + a) + 175*B*a^4*b*x^2*sgn(b*x + a) + 350*A*a^3*b^2*x^2*sgn(b*x + a) + 21*B*a^5*x*sgn(b*x
+ a) + 105*A*a^4*b*x*sgn(b*x + a) + 15*A*a^5*sgn(b*x + a))/x^(7/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{9/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{x^{9/2}} \,d x \]

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/x^(9/2),x)

[Out]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/x^(9/2), x)