3.7.73 \(\int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx\) [673]

Optimal. Leaf size=280 \[ -\frac {5 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}{512 a^3 c^3 x}+\frac {5 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}{768 a^2 c^3 x^2}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac {5 (b c-a d)^6 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{7/2} c^{7/2}} \]

[Out]

-1/12*(-a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(7/2)/c^2/x^5-1/6*(b*x+a)^(5/2)*(d*x+c)^(7/2)/c/x^6+5/512*(-a*d+b*c)^6*
arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(7/2)+5/768*(-a*d+b*c)^4*(d*x+c)^(3/2)*(b*x+a)^
(1/2)/a^2/c^3/x^2-1/192*(-a*d+b*c)^3*(d*x+c)^(5/2)*(b*x+a)^(1/2)/a/c^3/x^3-1/32*(-a*d+b*c)^2*(d*x+c)^(7/2)*(b*
x+a)^(1/2)/c^3/x^4-5/512*(-a*d+b*c)^5*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3/x

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Rubi [A]
time = 0.12, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214} \begin {gather*} \frac {5 (b c-a d)^6 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{7/2} c^{7/2}}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^5}{512 a^3 c^3 x}+\frac {5 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^4}{768 a^2 c^3 x^2}-\frac {\sqrt {a+b x} (c+d x)^{7/2} (b c-a d)^2}{32 c^3 x^4}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (b c-a d)^3}{192 a c^3 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{7/2} (b c-a d)}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7,x]

[Out]

(-5*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*a^3*c^3*x) + (5*(b*c - a*d)^4*Sqrt[a + b*x]*(c + d*x)^(3/2
))/(768*a^2*c^3*x^2) - ((b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(5/2))/(192*a*c^3*x^3) - ((b*c - a*d)^2*Sqrt[a +
b*x]*(c + d*x)^(7/2))/(32*c^3*x^4) - ((b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(12*c^2*x^5) - ((a + b*x)^
(5/2)*(c + d*x)^(7/2))/(6*c*x^6) + (5*(b*c - a*d)^6*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/
(512*a^(7/2)*c^(7/2))

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 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 {(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac {(5 (b c-a d)) \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx}{12 c}\\ &=-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac {(b c-a d)^2 \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^5} \, dx}{8 c^2}\\ &=-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac {(b c-a d)^3 \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx}{64 c^3}\\ &=-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}-\frac {\left (5 (b c-a d)^4\right ) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx}{384 a c^3}\\ &=\frac {5 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}{768 a^2 c^3 x^2}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac {\left (5 (b c-a d)^5\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{512 a^2 c^3}\\ &=-\frac {5 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}{512 a^3 c^3 x}+\frac {5 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}{768 a^2 c^3 x^2}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}-\frac {\left (5 (b c-a d)^6\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{1024 a^3 c^3}\\ &=-\frac {5 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}{512 a^3 c^3 x}+\frac {5 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}{768 a^2 c^3 x^2}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}-\frac {\left (5 (b c-a d)^6\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{512 a^3 c^3}\\ &=-\frac {5 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}{512 a^3 c^3 x}+\frac {5 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}{768 a^2 c^3 x^2}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac {5 (b c-a d)^6 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{7/2} c^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.62, size = 206, normalized size = 0.74 \begin {gather*} \frac {(b c-a d)^6 \left (-\frac {\sqrt {a} \sqrt {c} (a+b x)^{11/2} \sqrt {c+d x} \left (15 c^5-\frac {85 a c^4 (c+d x)}{a+b x}+\frac {198 a^2 c^3 (c+d x)^2}{(a+b x)^2}+\frac {198 a^3 c^2 (c+d x)^3}{(a+b x)^3}-\frac {85 a^4 c (c+d x)^4}{(a+b x)^4}+\frac {15 a^5 (c+d x)^5}{(a+b x)^5}\right )}{(b c x-a d x)^6}+15 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )}{1536 a^{7/2} c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7,x]

[Out]

((b*c - a*d)^6*(-((Sqrt[a]*Sqrt[c]*(a + b*x)^(11/2)*Sqrt[c + d*x]*(15*c^5 - (85*a*c^4*(c + d*x))/(a + b*x) + (
198*a^2*c^3*(c + d*x)^2)/(a + b*x)^2 + (198*a^3*c^2*(c + d*x)^3)/(a + b*x)^3 - (85*a^4*c*(c + d*x)^4)/(a + b*x
)^4 + (15*a^5*(c + d*x)^5)/(a + b*x)^5))/(b*c*x - a*d*x)^6) + 15*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt
[a + b*x])]))/(1536*a^(7/2)*c^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1067\) vs. \(2(230)=460\).
time = 0.09, size = 1068, normalized size = 3.81

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (170 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b c \,d^{4} x^{5}-396 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c^{2} d^{3} x^{5}-396 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{3} d^{2} x^{5}+170 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{4} c^{4} d \,x^{5}-90 \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} b c \,d^{5} x^{6}+225 \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^{2} c^{2} d^{4} x^{6}-300 \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^{3} c^{3} d^{3} x^{6}+225 \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^{4} c^{4} d^{2} x^{6}-90 \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^{5} c^{5} d \,x^{6}-16 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} c^{2} d^{3} x^{3}-16 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{5} x^{3}-864 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} c^{3} d^{2} x^{2}-1280 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} c^{4} d x -1280 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b \,c^{5} x +20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} c \,d^{4} x^{4}+20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{4} c^{5} x^{4}-864 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c^{5} x^{2}-512 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} c^{5} \sqrt {a c}+15 \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^{6} d^{6} x^{6}+15 \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^{6} c^{6} x^{6}-112 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b \,c^{2} d^{3} x^{4}-2376 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c^{3} d^{2} x^{4}-112 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{4} d \,x^{4}-2544 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b \,c^{3} d^{2} x^{3}-2544 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c^{4} d \,x^{3}-3392 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b \,c^{4} d \,x^{2}-30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{5} c^{5} x^{5}-30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} d^{5} x^{5}\right )}{3072 a^{3} c^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{6} \sqrt {a c}}\)	\(1068\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

1/3072*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3*(170*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b*c*d^4*x^5-396*(a*c)^
(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b^2*c^2*d^3*x^5-396*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^3*c^3*d^2*x^5+
170*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^4*c^4*d*x^5-90*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*b*c*d^5*x^6+225*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^2*c^2*d
^4*x^6-300*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^3*c^3*d^3*x^6+225*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^4*c^4*d^2*x^6-90*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^5*c^5*d*x^6-16*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^5*c^2*d^3*x^3-16*(a*c)^
(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^3*c^5*x^3-864*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^5*c^3*d^2*x^2-1280*(a*
c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^5*c^4*d*x-1280*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b*c^5*x+20*(a*c)^(1/
2)*((d*x+c)*(b*x+a))^(1/2)*a^5*c*d^4*x^4+20*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^4*c^5*x^4-864*(a*c)^(1/2)*
((d*x+c)*(b*x+a))^(1/2)*a^3*b^2*c^5*x^2-512*((d*x+c)*(b*x+a))^(1/2)*a^5*c^5*(a*c)^(1/2)+15*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^6*d^6*x^6+15*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^6*c^6*x^6-112*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b*c^2*d^3*x^4-2376*(a*c)^(1/2)*((d*x+c)*
(b*x+a))^(1/2)*a^3*b^2*c^3*d^2*x^4-112*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^3*c^4*d*x^4-2544*(a*c)^(1/2)*
((d*x+c)*(b*x+a))^(1/2)*a^4*b*c^3*d^2*x^3-2544*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b^2*c^4*d*x^3-3392*(a*c
)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b*c^4*d*x^2-30*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^5*c^5*x^5-30*(a*c)^(1
/2)*((d*x+c)*(b*x+a))^(1/2)*a^5*d^5*x^5)/((d*x+c)*(b*x+a))^(1/2)/x^6/(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((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,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 = 10.51, size = 908, normalized size = 3.24 \begin {gather*} \left [\frac {15 \, {\left (b^{6} c^{6} - 6 \, a b^{5} c^{5} d + 15 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 15 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + a^{6} d^{6}\right )} \sqrt {a c} x^{6} \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 (256 \, a^{6} c^{6} + {\left (15 \, a b^{5} c^{6} - 85 \, a^{2} b^{4} c^{5} d + 198 \, a^{3} b^{3} c^{4} d^{2} + 198 \, a^{4} b^{2} c^{3} d^{3} - 85 \, a^{5} b c^{2} d^{4} + 15 \, a^{6} c d^{5}\right )} x^{5} - 2 \, {\left (5 \, a^{2} b^{4} c^{6} - 28 \, a^{3} b^{3} c^{5} d - 594 \, a^{4} b^{2} c^{4} d^{2} - 28 \, a^{5} b c^{3} d^{3} + 5 \, a^{6} c^{2} d^{4}\right )} x^{4} + 8 \, {\left (a^{3} b^{3} c^{6} + 159 \, a^{4} b^{2} c^{5} d + 159 \, a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{3} + 16 \, {\left (27 \, a^{4} b^{2} c^{6} + 106 \, a^{5} b c^{5} d + 27 \, a^{6} c^{4} d^{2}\right )} x^{2} + 640 \, {\left (a^{5} b c^{6} + a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6144 \, a^{4} c^{4} x^{6}}, -\frac {15 \, {\left (b^{6} c^{6} - 6 \, a b^{5} c^{5} d + 15 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 15 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + a^{6} d^{6}\right )} \sqrt {-a c} x^{6} \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 (256 \, a^{6} c^{6} + {\left (15 \, a b^{5} c^{6} - 85 \, a^{2} b^{4} c^{5} d + 198 \, a^{3} b^{3} c^{4} d^{2} + 198 \, a^{4} b^{2} c^{3} d^{3} - 85 \, a^{5} b c^{2} d^{4} + 15 \, a^{6} c d^{5}\right )} x^{5} - 2 \, {\left (5 \, a^{2} b^{4} c^{6} - 28 \, a^{3} b^{3} c^{5} d - 594 \, a^{4} b^{2} c^{4} d^{2} - 28 \, a^{5} b c^{3} d^{3} + 5 \, a^{6} c^{2} d^{4}\right )} x^{4} + 8 \, {\left (a^{3} b^{3} c^{6} + 159 \, a^{4} b^{2} c^{5} d + 159 \, a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{3} + 16 \, {\left (27 \, a^{4} b^{2} c^{6} + 106 \, a^{5} b c^{5} d + 27 \, a^{6} c^{4} d^{2}\right )} x^{2} + 640 \, {\left (a^{5} b c^{6} + a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3072 \, a^{4} c^{4} x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,x, algorithm="fricas")

[Out]

[1/6144*(15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*
c*d^5 + a^6*d^6)*sqrt(a*c)*x^6*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*(256*a^6*c^6 + (15*a*b^5*c^6 - 85*
a^2*b^4*c^5*d + 198*a^3*b^3*c^4*d^2 + 198*a^4*b^2*c^3*d^3 - 85*a^5*b*c^2*d^4 + 15*a^6*c*d^5)*x^5 - 2*(5*a^2*b^
4*c^6 - 28*a^3*b^3*c^5*d - 594*a^4*b^2*c^4*d^2 - 28*a^5*b*c^3*d^3 + 5*a^6*c^2*d^4)*x^4 + 8*(a^3*b^3*c^6 + 159*
a^4*b^2*c^5*d + 159*a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^3 + 16*(27*a^4*b^2*c^6 + 106*a^5*b*c^5*d + 27*a^6*c^4*d^2)*
x^2 + 640*(a^5*b*c^6 + a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^6), -1/3072*(15*(b^6*c^6 - 6*a*b^
5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*sqrt(-a*c)*x
^6*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*(256*a^6*c^6 + (15*a*b^5*c^6 - 85*a^2*b^4*c^5*d + 198*a^3*b^3*c^4*d^2 + 198*a^4*b^2*c^3*d^
3 - 85*a^5*b*c^2*d^4 + 15*a^6*c*d^5)*x^5 - 2*(5*a^2*b^4*c^6 - 28*a^3*b^3*c^5*d - 594*a^4*b^2*c^4*d^2 - 28*a^5*
b*c^3*d^3 + 5*a^6*c^2*d^4)*x^4 + 8*(a^3*b^3*c^6 + 159*a^4*b^2*c^5*d + 159*a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^3 + 1
6*(27*a^4*b^2*c^6 + 106*a^5*b*c^5*d + 27*a^6*c^4*d^2)*x^2 + 640*(a^5*b*c^6 + a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(
d*x + c))/(a^4*c^4*x^6)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(d*x+c)**(5/2)/x**7,x)

[Out]

Integral((a + b*x)**(5/2)*(c + d*x)**(5/2)/x**7, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 8500 vs. \(2 (230) = 460\).
time = 10.70, size = 8500, normalized size = 30.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,x, algorithm="giac")

[Out]

1/1536*(15*(sqrt(b*d)*b^7*c^6*abs(b) - 6*sqrt(b*d)*a*b^6*c^5*d*abs(b) + 15*sqrt(b*d)*a^2*b^5*c^4*d^2*abs(b) -
20*sqrt(b*d)*a^3*b^4*c^3*d^3*abs(b) + 15*sqrt(b*d)*a^4*b^3*c^2*d^4*abs(b) - 6*sqrt(b*d)*a^5*b^2*c*d^5*abs(b) +
sqrt(b*d)*a^6*b*d^6*abs(b))*arctan(-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^3*b*c^3) - 2*(15*sqrt(b*d)*b^29*c^17*abs(b) - 265*sqrt(b*
d)*a*b^28*c^16*d*abs(b) + 2208*sqrt(b*d)*a^2*b^27*c^15*d^2*abs(b) - 11088*sqrt(b*d)*a^3*b^26*c^14*d^3*abs(b) +
36732*sqrt(b*d)*a^4*b^25*c^13*d^4*abs(b) - 83412*sqrt(b*d)*a^5*b^24*c^12*d^5*abs(b) + 129840*sqrt(b*d)*a^6*b^
23*c^11*d^6*abs(b) - 129536*sqrt(b*d)*a^7*b^22*c^10*d^7*abs(b) + 55506*sqrt(b*d)*a^8*b^21*c^9*d^8*abs(b) + 555
06*sqrt(b*d)*a^9*b^20*c^8*d^9*abs(b) - 129536*sqrt(b*d)*a^10*b^19*c^7*d^10*abs(b) + 129840*sqrt(b*d)*a^11*b^18
*c^6*d^11*abs(b) - 83412*sqrt(b*d)*a^12*b^17*c^5*d^12*abs(b) + 36732*sqrt(b*d)*a^13*b^16*c^4*d^13*abs(b) - 110
88*sqrt(b*d)*a^14*b^15*c^3*d^14*abs(b) + 2208*sqrt(b*d)*a^15*b^14*c^2*d^15*abs(b) - 265*sqrt(b*d)*a^16*b^13*c*
d^16*abs(b) + 15*sqrt(b*d)*a^17*b^12*d^17*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^2*b^27*c^16*abs(b) + 2400*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^26*c^15*d*abs(b) - 16056*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^25*c^14*d^2*abs(b) + 62880*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^24*c^13*d^3*abs(b) - 156876*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^23*c^12*d^4*abs(b) + 257760*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^22*c^11*d^5*abs(b) - 282120*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^21*c^10*d^6*abs(b) + 217632*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^20*c^9*d^7*abs(b) - 170910*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^19*c^8*d^8*abs(b) + 217632*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^2*a^9*b^18*c^7*d^9*abs(b) - 282120*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
2*a^10*b^17*c^6*d^10*abs(b) + 257760*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^2*a^11*b^16*c^5*d^11*abs(b) - 156876*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^15*c^4*d^12*abs(b) + 62880*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^14*c^3*d^13*abs(b) - 16056*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^2*a^14*b^13*c^2*d^14*abs(b) + 2400*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^2*a^15*b^12*c*d^15*abs(b) - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a
^16*b^11*d^16*abs(b) + 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^25*c^
15*abs(b) - 9765*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^24*c^14*d*abs
(b) + 51813*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^23*c^13*d^2*abs(
b) - 147657*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^22*c^12*d^3*abs(
b) + 229845*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^21*c^11*d^4*abs(
b) - 168225*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^20*c^10*d^5*abs(
b) + 6945*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^19*c^9*d^6*abs(b)
+ 36219*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^18*c^8*d^7*abs(b) +
36219*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^17*c^7*d^8*abs(b) + 69
45*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^16*c^6*d^9*abs(b) - 16822
5*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^15*c^5*d^10*abs(b) + 2298
45*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^14*c^4*d^11*abs(b) - 147
657*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^13*c^3*d^12*abs(b) + 51
813*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^13*b^12*c^2*d^13*abs(b) - 97
65*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^14*b^11*c*d^14*abs(b) + 825*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^15*b^10*d^15*abs(b) - 2475*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^23*c^14*abs(b) + 23610*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^22*c^13*d*abs(b) - 98505*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7,x)

[Out]

int(((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7, x)

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