3.2216 \(\int \frac {x^3}{(a+b \sqrt {x})^5} \, dx\)

Optimal. Leaf size=131 \[ \frac {a^7}{2 b^8 \left (a+b \sqrt {x}\right )^4}-\frac {14 a^6}{3 b^8 \left (a+b \sqrt {x}\right )^3}+\frac {21 a^5}{b^8 \left (a+b \sqrt {x}\right )^2}-\frac {70 a^4}{b^8 \left (a+b \sqrt {x}\right )}-\frac {70 a^3 \log \left (a+b \sqrt {x}\right )}{b^8}+\frac {30 a^2 \sqrt {x}}{b^7}-\frac {5 a x}{b^6}+\frac {2 x^{3/2}}{3 b^5} \]

[Out]

-5*a*x/b^6+2/3*x^(3/2)/b^5-70*a^3*ln(a+b*x^(1/2))/b^8+30*a^2*x^(1/2)/b^7+1/2*a^7/b^8/(a+b*x^(1/2))^4-14/3*a^6/
b^8/(a+b*x^(1/2))^3+21*a^5/b^8/(a+b*x^(1/2))^2-70*a^4/b^8/(a+b*x^(1/2))

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Rubi [A]  time = 0.10, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {a^7}{2 b^8 \left (a+b \sqrt {x}\right )^4}-\frac {14 a^6}{3 b^8 \left (a+b \sqrt {x}\right )^3}+\frac {21 a^5}{b^8 \left (a+b \sqrt {x}\right )^2}-\frac {70 a^4}{b^8 \left (a+b \sqrt {x}\right )}+\frac {30 a^2 \sqrt {x}}{b^7}-\frac {70 a^3 \log \left (a+b \sqrt {x}\right )}{b^8}-\frac {5 a x}{b^6}+\frac {2 x^{3/2}}{3 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Sqrt[x])^5,x]

[Out]

a^7/(2*b^8*(a + b*Sqrt[x])^4) - (14*a^6)/(3*b^8*(a + b*Sqrt[x])^3) + (21*a^5)/(b^8*(a + b*Sqrt[x])^2) - (70*a^
4)/(b^8*(a + b*Sqrt[x])) + (30*a^2*Sqrt[x])/b^7 - (5*a*x)/b^6 + (2*x^(3/2))/(3*b^5) - (70*a^3*Log[a + b*Sqrt[x
]])/b^8

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^3}{\left (a+b \sqrt {x}\right )^5} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^7}{(a+b x)^5} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {15 a^2}{b^7}-\frac {5 a x}{b^6}+\frac {x^2}{b^5}-\frac {a^7}{b^7 (a+b x)^5}+\frac {7 a^6}{b^7 (a+b x)^4}-\frac {21 a^5}{b^7 (a+b x)^3}+\frac {35 a^4}{b^7 (a+b x)^2}-\frac {35 a^3}{b^7 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^7}{2 b^8 \left (a+b \sqrt {x}\right )^4}-\frac {14 a^6}{3 b^8 \left (a+b \sqrt {x}\right )^3}+\frac {21 a^5}{b^8 \left (a+b \sqrt {x}\right )^2}-\frac {70 a^4}{b^8 \left (a+b \sqrt {x}\right )}+\frac {30 a^2 \sqrt {x}}{b^7}-\frac {5 a x}{b^6}+\frac {2 x^{3/2}}{3 b^5}-\frac {70 a^3 \log \left (a+b \sqrt {x}\right )}{b^8}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 126, normalized size = 0.96 \[ \frac {-319 a^7-856 a^6 b \sqrt {x}-444 a^5 b^2 x+544 a^4 b^3 x^{3/2}+556 a^3 b^4 x^2-420 a^3 \left (a+b \sqrt {x}\right )^4 \log \left (a+b \sqrt {x}\right )+84 a^2 b^5 x^{5/2}-14 a b^6 x^3+4 b^7 x^{7/2}}{6 b^8 \left (a+b \sqrt {x}\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Sqrt[x])^5,x]

[Out]

(-319*a^7 - 856*a^6*b*Sqrt[x] - 444*a^5*b^2*x + 544*a^4*b^3*x^(3/2) + 556*a^3*b^4*x^2 + 84*a^2*b^5*x^(5/2) - 1
4*a*b^6*x^3 + 4*b^7*x^(7/2) - 420*a^3*(a + b*Sqrt[x])^4*Log[a + b*Sqrt[x]])/(6*b^8*(a + b*Sqrt[x])^4)

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fricas [B]  time = 0.67, size = 223, normalized size = 1.70 \[ -\frac {30 \, a b^{10} x^{5} - 120 \, a^{3} b^{8} x^{4} - 366 \, a^{5} b^{6} x^{3} + 1179 \, a^{7} b^{4} x^{2} - 1066 \, a^{9} b^{2} x + 319 \, a^{11} + 420 \, {\left (a^{3} b^{8} x^{4} - 4 \, a^{5} b^{6} x^{3} + 6 \, a^{7} b^{4} x^{2} - 4 \, a^{9} b^{2} x + a^{11}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (b^{11} x^{5} + 41 \, a^{2} b^{9} x^{4} - 279 \, a^{4} b^{7} x^{3} + 511 \, a^{6} b^{5} x^{2} - 385 \, a^{8} b^{3} x + 105 \, a^{10} b\right )} \sqrt {x}}{6 \, {\left (b^{16} x^{4} - 4 \, a^{2} b^{14} x^{3} + 6 \, a^{4} b^{12} x^{2} - 4 \, a^{6} b^{10} x + a^{8} b^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

-1/6*(30*a*b^10*x^5 - 120*a^3*b^8*x^4 - 366*a^5*b^6*x^3 + 1179*a^7*b^4*x^2 - 1066*a^9*b^2*x + 319*a^11 + 420*(
a^3*b^8*x^4 - 4*a^5*b^6*x^3 + 6*a^7*b^4*x^2 - 4*a^9*b^2*x + a^11)*log(b*sqrt(x) + a) - 4*(b^11*x^5 + 41*a^2*b^
9*x^4 - 279*a^4*b^7*x^3 + 511*a^6*b^5*x^2 - 385*a^8*b^3*x + 105*a^10*b)*sqrt(x))/(b^16*x^4 - 4*a^2*b^14*x^3 +
6*a^4*b^12*x^2 - 4*a^6*b^10*x + a^8*b^8)

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giac [A]  time = 0.16, size = 99, normalized size = 0.76 \[ -\frac {70 \, a^{3} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{8}} - \frac {420 \, a^{4} b^{3} x^{\frac {3}{2}} + 1134 \, a^{5} b^{2} x + 1036 \, a^{6} b \sqrt {x} + 319 \, a^{7}}{6 \, {\left (b \sqrt {x} + a\right )}^{4} b^{8}} + \frac {2 \, b^{10} x^{\frac {3}{2}} - 15 \, a b^{9} x + 90 \, a^{2} b^{8} \sqrt {x}}{3 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

-70*a^3*log(abs(b*sqrt(x) + a))/b^8 - 1/6*(420*a^4*b^3*x^(3/2) + 1134*a^5*b^2*x + 1036*a^6*b*sqrt(x) + 319*a^7
)/((b*sqrt(x) + a)^4*b^8) + 1/3*(2*b^10*x^(3/2) - 15*a*b^9*x + 90*a^2*b^8*sqrt(x))/b^15

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maple [A]  time = 0.01, size = 112, normalized size = 0.85 \[ \frac {a^{7}}{2 \left (b \sqrt {x}+a \right )^{4} b^{8}}-\frac {14 a^{6}}{3 \left (b \sqrt {x}+a \right )^{3} b^{8}}+\frac {21 a^{5}}{\left (b \sqrt {x}+a \right )^{2} b^{8}}+\frac {2 x^{\frac {3}{2}}}{3 b^{5}}-\frac {70 a^{4}}{\left (b \sqrt {x}+a \right ) b^{8}}-\frac {70 a^{3} \ln \left (b \sqrt {x}+a \right )}{b^{8}}-\frac {5 a x}{b^{6}}+\frac {30 a^{2} \sqrt {x}}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^(1/2)+a)^5,x)

[Out]

-5*a*x/b^6+2/3*x^(3/2)/b^5-70*a^3*ln(b*x^(1/2)+a)/b^8+30*a^2*x^(1/2)/b^7+1/2*a^7/b^8/(b*x^(1/2)+a)^4-14/3*a^6/
b^8/(b*x^(1/2)+a)^3+21*a^5/b^8/(b*x^(1/2)+a)^2-70*a^4/b^8/(b*x^(1/2)+a)

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maxima [A]  time = 0.90, size = 129, normalized size = 0.98 \[ -\frac {70 \, a^{3} \log \left (b \sqrt {x} + a\right )}{b^{8}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{3}}{3 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{2} a}{b^{8}} + \frac {42 \, {\left (b \sqrt {x} + a\right )} a^{2}}{b^{8}} - \frac {70 \, a^{4}}{{\left (b \sqrt {x} + a\right )} b^{8}} + \frac {21 \, a^{5}}{{\left (b \sqrt {x} + a\right )}^{2} b^{8}} - \frac {14 \, a^{6}}{3 \, {\left (b \sqrt {x} + a\right )}^{3} b^{8}} + \frac {a^{7}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

-70*a^3*log(b*sqrt(x) + a)/b^8 + 2/3*(b*sqrt(x) + a)^3/b^8 - 7*(b*sqrt(x) + a)^2*a/b^8 + 42*(b*sqrt(x) + a)*a^
2/b^8 - 70*a^4/((b*sqrt(x) + a)*b^8) + 21*a^5/((b*sqrt(x) + a)^2*b^8) - 14/3*a^6/((b*sqrt(x) + a)^3*b^8) + 1/2
*a^7/((b*sqrt(x) + a)^4*b^8)

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mupad [B]  time = 1.20, size = 126, normalized size = 0.96 \[ \frac {2\,x^{3/2}}{3\,b^5}-\frac {\frac {319\,a^7}{6\,b}+\frac {518\,a^6\,\sqrt {x}}{3}+70\,a^4\,b^2\,x^{3/2}+189\,a^5\,b\,x}{a^4\,b^7+b^{11}\,x^2+6\,a^2\,b^9\,x+4\,a\,b^{10}\,x^{3/2}+4\,a^3\,b^8\,\sqrt {x}}-\frac {70\,a^3\,\ln \left (a+b\,\sqrt {x}\right )}{b^8}+\frac {30\,a^2\,\sqrt {x}}{b^7}-\frac {5\,a\,x}{b^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^(1/2))^5,x)

[Out]

(2*x^(3/2))/(3*b^5) - ((319*a^7)/(6*b) + (518*a^6*x^(1/2))/3 + 70*a^4*b^2*x^(3/2) + 189*a^5*b*x)/(a^4*b^7 + b^
11*x^2 + 6*a^2*b^9*x + 4*a*b^10*x^(3/2) + 4*a^3*b^8*x^(1/2)) - (70*a^3*log(a + b*x^(1/2)))/b^8 + (30*a^2*x^(1/
2))/b^7 - (5*a*x)/b^6

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sympy [A]  time = 3.03, size = 818, normalized size = 6.24 \[ \begin {cases} - \frac {420 a^{7} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} - \frac {875 a^{7}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} - \frac {1680 a^{6} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} - \frac {3080 a^{6} b \sqrt {x}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} - \frac {2520 a^{5} b^{2} x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} - \frac {3780 a^{5} b^{2} x}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} - \frac {1680 a^{4} b^{3} x^{\frac {3}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} - \frac {1680 a^{4} b^{3} x^{\frac {3}{2}}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} - \frac {420 a^{3} b^{4} x^{2} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} + \frac {84 a^{2} b^{5} x^{\frac {5}{2}}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} - \frac {14 a b^{6} x^{3}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} + \frac {4 b^{7} x^{\frac {7}{2}}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt {x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac {3}{2}} + 6 b^{12} x^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{5}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+b*x**(1/2))**5,x)

[Out]

Piecewise((-420*a**7*log(a/b + sqrt(x))/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**
(3/2) + 6*b**12*x**2) - 875*a**7/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) +
6*b**12*x**2) - 1680*a**6*b*sqrt(x)*log(a/b + sqrt(x))/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x
+ 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 3080*a**6*b*sqrt(x)/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**1
0*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 2520*a**5*b**2*x*log(a/b + sqrt(x))/(6*a**4*b**8 + 24*a**3*b**9*sq
rt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 3780*a**5*b**2*x/(6*a**4*b**8 + 24*a**3*b**9*s
qrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 1680*a**4*b**3*x**(3/2)*log(a/b + sqrt(x))/(6
*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 1680*a**4*b**3*x**
(3/2)/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 420*a**3*b
**4*x**2*log(a/b + sqrt(x))/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b*
*12*x**2) + 84*a**2*b**5*x**(5/2)/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2)
+ 6*b**12*x**2) - 14*a*b**6*x**3/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) +
6*b**12*x**2) + 4*b**7*x**(7/2)/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) +
6*b**12*x**2), Ne(b, 0)), (x**4/(4*a**5), True))

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