3.6.74 \(\int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx\) [574]

Optimal. Leaf size=283 \[ \frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^4 c^2 x}-\frac {(b c-a d)^2 (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac {(b c-a d)^4 (7 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{5/2}} \]

[Out]

-1/5*(b*x+a)^(3/2)*(d*x+c)^(7/2)/a/c/x^5-1/128*(-a*d+b*c)^4*(3*a*d+7*b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2
)/(d*x+c)^(1/2))/a^(9/2)/c^(5/2)-1/192*(-a*d+b*c)^2*(3*a*d+7*b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a^3/c^2/x^2+1/24
0*(-a*d+b*c)*(3*a*d+7*b*c)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/a^2/c^2/x^3+1/40*(3*a*d+7*b*c)*(d*x+c)^(7/2)*(b*x+a)^(1
/2)/a/c^2/x^4+1/128*(-a*d+b*c)^3*(3*a*d+7*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^2/x

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Rubi [A]
time = 0.11, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {98, 96, 95, 214} \begin {gather*} -\frac {(3 a d+7 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+7 b c) (b c-a d)^3}{128 a^4 c^2 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (3 a d+7 b c) (b c-a d)^2}{192 a^3 c^2 x^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (3 a d+7 b c) (b c-a d)}{240 a^2 c^2 x^3}+\frac {\sqrt {a+b x} (c+d x)^{7/2} (3 a d+7 b c)}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^6,x]

[Out]

((b*c - a*d)^3*(7*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^4*c^2*x) - ((b*c - a*d)^2*(7*b*c + 3*a*d)*S
qrt[a + b*x]*(c + d*x)^(3/2))/(192*a^3*c^2*x^2) + ((b*c - a*d)*(7*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/
(240*a^2*c^2*x^3) + ((7*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/(40*a*c^2*x^4) - ((a + b*x)^(3/2)*(c + d*x
)^(7/2))/(5*a*c*x^5) - ((b*c - a*d)^4*(7*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]
)/(128*a^(9/2)*c^(5/2))

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx &=-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac {\left (\frac {7 b c}{2}+\frac {3 a d}{2}\right ) \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^5} \, dx}{5 a c}\\ &=\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac {((b c-a d) (7 b c+3 a d)) \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx}{80 a c^2}\\ &=\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}+\frac {\left ((b c-a d)^2 (7 b c+3 a d)\right ) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx}{96 a^2 c^2}\\ &=-\frac {(b c-a d)^2 (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac {\left ((b c-a d)^3 (7 b c+3 a d)\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{128 a^3 c^2}\\ &=\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^4 c^2 x}-\frac {(b c-a d)^2 (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}+\frac {\left ((b c-a d)^4 (7 b c+3 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^4 c^2}\\ &=\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^4 c^2 x}-\frac {(b c-a d)^2 (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}+\frac {\left ((b c-a d)^4 (7 b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 a^4 c^2}\\ &=\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^4 c^2 x}-\frac {(b c-a d)^2 (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac {(b c-a d)^4 (7 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.49, size = 244, normalized size = 0.86 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4 x^4-10 a b^3 c^3 x^3 (7 c+34 d x)+2 a^2 b^2 c^2 x^2 \left (28 c^2+111 c d x+173 d^2 x^2\right )-2 a^3 b c x \left (24 c^3+88 c^2 d x+109 c d^2 x^2+30 d^3 x^3\right )-3 a^4 \left (128 c^4+336 c^3 d x+248 c^2 d^2 x^2+10 c d^3 x^3-15 d^4 x^4\right )\right )}{1920 a^4 c^2 x^5}-\frac {(b c-a d)^4 (7 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^6,x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(105*b^4*c^4*x^4 - 10*a*b^3*c^3*x^3*(7*c + 34*d*x) + 2*a^2*b^2*c^2*x^2*(28*c^2 +
111*c*d*x + 173*d^2*x^2) - 2*a^3*b*c*x*(24*c^3 + 88*c^2*d*x + 109*c*d^2*x^2 + 30*d^3*x^3) - 3*a^4*(128*c^4 + 3
36*c^3*d*x + 248*c^2*d^2*x^2 + 10*c*d^3*x^3 - 15*d^4*x^4)))/(1920*a^4*c^2*x^5) - ((b*c - a*d)^4*(7*b*c + 3*a*d
)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(9/2)*c^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(812\) vs. \(2(239)=478\).
time = 0.06, size = 813, normalized size = 2.87

method	result	size
default	\(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{5} d^{5} x^{5}-75 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} b c \,d^{4} x^{5}-150 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b^{2} c^{2} d^{3} x^{5}+450 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{3} c^{3} d^{2} x^{5}-375 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{4} c^{4} d \,x^{5}+105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{5} c^{5} x^{5}-90 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} d^{4} x^{4}+120 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b c \,d^{3} x^{4}-692 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{2} d^{2} x^{4}+680 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{3} d \,x^{4}-210 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{4} c^{4} x^{4}+60 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c \,d^{3} x^{3}+436 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{2} d^{2} x^{3}-444 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{3} d \,x^{3}+140 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{4} x^{3}+1488 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{2} d^{2} x^{2}+352 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{3} d \,x^{2}-112 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{4} x^{2}+2016 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{3} d x +96 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{4} x +768 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{4} \sqrt {a c}\right )}{3840 a^{4} c^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{5} \sqrt {a c}}\)	\(813\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/3840*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^4/c^2*(45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x
)*a^5*d^5*x^5-75*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^4*b*c*d^4*x^5-150*ln((a*d*x
+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*b^2*c^2*d^3*x^5+450*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(
(d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b^3*c^3*d^2*x^5-375*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)
+2*a*c)/x)*a*b^4*c^4*d*x^5+105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*b^5*c^5*x^5-90*
(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*d^4*x^4+120*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c*d^3*x^4-692*(a
*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^2*d^2*x^4+680*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^3*c^3*d*x^4-
210*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^4*c^4*x^4+60*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c*d^3*x^3+436*(
a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c^2*d^2*x^3-444*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^3*d*x^3
+140*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^3*c^4*x^3+1488*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c^2*d^2*x^
2+352*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c^3*d*x^2-112*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^4*
x^2+2016*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c^3*d*x+96*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c^4*x+76
8*((d*x+c)*(b*x+a))^(1/2)*a^4*c^4*(a*c)^(1/2))/((d*x+c)*(b*x+a))^(1/2)/x^5/(a*c)^(1/2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [A]
time = 5.03, size = 732, normalized size = 2.59 \begin {gather*} \left [\frac {15 \, {\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5}\right )} \sqrt {a c} x^{5} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 340 \, a^{2} b^{3} c^{4} d + 346 \, a^{3} b^{2} c^{3} d^{2} - 60 \, a^{4} b c^{2} d^{3} + 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 111 \, a^{3} b^{2} c^{4} d + 109 \, a^{4} b c^{3} d^{2} + 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 22 \, a^{4} b c^{4} d - 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + 21 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{5} c^{3} x^{5}}, \frac {15 \, {\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 340 \, a^{2} b^{3} c^{4} d + 346 \, a^{3} b^{2} c^{3} d^{2} - 60 \, a^{4} b c^{2} d^{3} + 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 111 \, a^{3} b^{2} c^{4} d + 109 \, a^{4} b c^{3} d^{2} + 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 22 \, a^{4} b c^{4} d - 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + 21 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{5} c^{3} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^6,x, algorithm="fricas")

[Out]

[1/7680*(15*(7*b^5*c^5 - 25*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 3*a^5*d^5)
*sqrt(a*c)*x^5*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt
(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(384*a^5*c^5 - (105*a*b^4*c^5 - 340*a^2*b^3*c^4*d
+ 346*a^3*b^2*c^3*d^2 - 60*a^4*b*c^2*d^3 + 45*a^5*c*d^4)*x^4 + 2*(35*a^2*b^3*c^5 - 111*a^3*b^2*c^4*d + 109*a^4
*b*c^3*d^2 + 15*a^5*c^2*d^3)*x^3 - 8*(7*a^3*b^2*c^5 - 22*a^4*b*c^4*d - 93*a^5*c^3*d^2)*x^2 + 48*(a^4*b*c^5 + 2
1*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^3*x^5), 1/3840*(15*(7*b^5*c^5 - 25*a*b^4*c^4*d + 30*a^2*b^
3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(-a*c)*x^5*arctan(1/2*(2*a*c + (b*c + a*d)*x)*
sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) - 2*(384*a^5*c^5 - (10
5*a*b^4*c^5 - 340*a^2*b^3*c^4*d + 346*a^3*b^2*c^3*d^2 - 60*a^4*b*c^2*d^3 + 45*a^5*c*d^4)*x^4 + 2*(35*a^2*b^3*c
^5 - 111*a^3*b^2*c^4*d + 109*a^4*b*c^3*d^2 + 15*a^5*c^2*d^3)*x^3 - 8*(7*a^3*b^2*c^5 - 22*a^4*b*c^4*d - 93*a^5*
c^3*d^2)*x^2 + 48*(a^4*b*c^5 + 21*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^3*x^5)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**6,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 5928 vs. \(2 (239) = 478\).
time = 28.82, size = 5928, normalized size = 20.95 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^6,x, algorithm="giac")

[Out]

-1/1920*(15*(7*sqrt(b*d)*b^6*c^5*abs(b) - 25*sqrt(b*d)*a*b^5*c^4*d*abs(b) + 30*sqrt(b*d)*a^2*b^4*c^3*d^2*abs(b
) - 10*sqrt(b*d)*a^3*b^3*c^2*d^3*abs(b) - 5*sqrt(b*d)*a^4*b^2*c*d^4*abs(b) + 3*sqrt(b*d)*a^5*b*d^5*abs(b))*arc
tan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b
))/(sqrt(-a*b*c*d)*a^4*b*c^2) - 2*(105*sqrt(b*d)*b^24*c^14*abs(b) - 1390*sqrt(b*d)*a*b^23*c^13*d*abs(b) + 8471
*sqrt(b*d)*a^2*b^22*c^12*d^2*abs(b) - 31420*sqrt(b*d)*a^3*b^21*c^11*d^3*abs(b) + 79065*sqrt(b*d)*a^4*b^20*c^10
*d^4*abs(b) - 142530*sqrt(b*d)*a^5*b^19*c^9*d^5*abs(b) + 189615*sqrt(b*d)*a^6*b^18*c^8*d^6*abs(b) - 189192*sqr
t(b*d)*a^7*b^17*c^7*d^7*abs(b) + 142755*sqrt(b*d)*a^8*b^16*c^6*d^8*abs(b) - 81810*sqrt(b*d)*a^9*b^15*c^5*d^9*a
bs(b) + 35725*sqrt(b*d)*a^10*b^14*c^4*d^10*abs(b) - 11900*sqrt(b*d)*a^11*b^13*c^3*d^11*abs(b) + 2971*sqrt(b*d)
*a^12*b^12*c^2*d^12*abs(b) - 510*sqrt(b*d)*a^13*b^11*c*d^13*abs(b) + 45*sqrt(b*d)*a^14*b^10*d^14*abs(b) - 945*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^22*c^13*abs(b) + 9535*sqrt(b*d)*
(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^21*c^12*d*abs(b) - 41870*sqrt(b*d)*(sqrt
(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^20*c^11*d^2*abs(b) + 102090*sqrt(b*d)*(sqrt
(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^19*c^10*d^3*abs(b) - 141675*sqrt(b*d)*(sqrt
(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^18*c^9*d^4*abs(b) + 84045*sqrt(b*d)*(sqrt(b
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^17*c^8*d^5*abs(b) + 64140*sqrt(b*d)*(sqrt(b*d
)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^16*c^7*d^6*abs(b) - 185220*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^15*c^6*d^7*abs(b) + 190665*sqrt(b*d)*(sqrt(b*d)*
sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^8*b^14*c^5*d^8*abs(b) - 117975*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^9*b^13*c^4*d^9*abs(b) + 49250*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^10*b^12*c^3*d^10*abs(b) - 14950*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^11*b^11*c^2*d^11*abs(b) + 3315*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^12*b^10*c*d^12*abs(b) - 405*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^13*b^9*d^13*abs(b) + 3780*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^20*c^12*abs(b) - 28200*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^
2*c + (b*x + a)*b*d - a*b*d))^4*a*b^19*c^11*d*abs(b) + 85640*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^4*a^2*b^18*c^10*d^2*abs(b) - 129800*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^4*a^3*b^17*c^9*d^3*abs(b) + 91020*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^4*a^4*b^16*c^8*d^4*abs(b) - 14800*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^4*a^5*b^15*c^7*d^5*abs(b) + 17200*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^4*a^6*b^14*c^6*d^6*abs(b) - 81360*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^4*a^7*b^13*c^5*d^7*abs(b) + 100700*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a
)*b*d - a*b*d))^4*a^8*b^12*c^4*d^8*abs(b) - 64520*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*
b*d - a*b*d))^4*a^9*b^11*c^3*d^9*abs(b) + 27720*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^4*a^10*b^10*c^2*d^10*abs(b) - 9000*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^4*a^11*b^9*c*d^11*abs(b) + 1620*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*a^12*b^8*d^12*abs(b) - 8820*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^6*b^18*c^11*abs(b) + 47180*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^17
*c^10*d*abs(b) - 95060*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^16*c^
9*d^2*abs(b) + 83340*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^15*c^8*
d^3*abs(b) - 39080*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^14*c^7*d^
4*abs(b) + 56280*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^13*c^6*d^5*
abs(b) - 77320*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6*b^12*c^5*d^6*ab
s(b) + 26040*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^7*b^11*c^4*d^7*abs(
b) + 22940*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^8*b^10*c^3*d^8*abs(b)
- 24740*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^9*b^9*c^2*d^9*abs(b) +
13020*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(1/2)*(c + d*x)^(5/2))/x^6,x)

[Out]

int(((a + b*x)^(1/2)*(c + d*x)^(5/2))/x^6, x)

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