3.7.64 \(\int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^6} \, dx\) [664]

Optimal. Leaf size=239 \[ \frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^3 x}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}-\frac {3 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{7/2}} \]

[Out]

-1/8*(-a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(5/2)/c^2/x^4-1/5*(b*x+a)^(5/2)*(d*x+c)^(5/2)/c/x^5-3/128*(-a*d+b*c)^5*a
rctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(5/2)/c^(7/2)-1/64*(-a*d+b*c)^3*(d*x+c)^(3/2)*(b*x+a)^(1
/2)/a/c^3/x^2-1/16*(-a*d+b*c)^2*(d*x+c)^(5/2)*(b*x+a)^(1/2)/c^3/x^3+3/128*(-a*d+b*c)^4*(b*x+a)^(1/2)*(d*x+c)^(
1/2)/a^2/c^3/x

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Rubi [A]
time = 0.09, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {3 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{7/2}}+\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^4}{128 a^2 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (b c-a d)^2}{16 c^3 x^3}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^3}{64 a c^3 x^2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (b c-a d)}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^6,x]

[Out]

(3*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^2*c^3*x) - ((b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(3/2))/
(64*a*c^3*x^2) - ((b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(5/2))/(16*c^3*x^3) - ((b*c - a*d)*(a + b*x)^(3/2)*(c
+ d*x)^(5/2))/(8*c^2*x^4) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(5*c*x^5) - (3*(b*c - a*d)^5*ArcTanh[(Sqrt[c]*Sq
rt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(5/2)*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)^{3/2}}{x^6} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac {(b c-a d) \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^5} \, dx}{2 c}\\ &=-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac {\left (3 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^4} \, dx}{16 c^2}\\ &=-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac {(b c-a d)^3 \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx}{32 c^3}\\ &=-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}-\frac {\left (3 (b c-a d)^4\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{128 a c^3}\\ &=\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^3 x}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac {\left (3 (b c-a d)^5\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^2 c^3}\\ &=\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^3 x}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac {\left (3 (b c-a d)^5\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^2 c^3}\\ &=\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^3 x}-\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac {(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}-\frac {3 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.53, size = 169, normalized size = 0.71 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 c^4 (a+b x)^4-70 a c^3 (a+b x)^3 (c+d x)-128 a^2 c^2 (a+b x)^2 (c+d x)^2+70 a^3 c (a+b x) (c+d x)^3-15 a^4 (c+d x)^4\right )}{640 a^2 c^3 x^5}+\frac {3 (-b c+a d)^5 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^6,x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*c^4*(a + b*x)^4 - 70*a*c^3*(a + b*x)^3*(c + d*x) - 128*a^2*c^2*(a + b*x)^2*(c
+ d*x)^2 + 70*a^3*c*(a + b*x)*(c + d*x)^3 - 15*a^4*(c + d*x)^4))/(640*a^2*c^3*x^5) + (3*(-(b*c) + a*d)^5*ArcT
anh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(5/2)*c^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(812\) vs. \(2(195)=390\).
time = 0.08, size = 813, normalized size = 3.40

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (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^{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}-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^{2} b^{3} c^{3} d^{2} 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 \,b^{4} c^{4} d \,x^{5}-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^{5} c^{5} x^{5}-30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} d^{4} x^{4}+140 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b c \,d^{3} x^{4}-256 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{2} d^{2} x^{4}-140 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{3} d \,x^{4}+30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{4} c^{4} x^{4}+20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c \,d^{3} x^{3}-92 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{2} d^{2} x^{3}-932 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{3} d \,x^{3}-20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{4} x^{3}-16 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{2} d^{2} x^{2}-1024 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{3} d \,x^{2}-496 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{4} x^{2}-352 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{3} d x -672 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{4} x -256 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{4} \sqrt {a c}\right )}{1280 a^{2} c^{3} \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((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

1/1280*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^3*(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^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^5-150*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^2*b^3*c^3*d^2*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*b^4*c^4*d*x^5-15*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^5*c^5*x^5-30*(a*
c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*d^4*x^4+140*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c*d^3*x^4-256*(a*c)
^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^2*d^2*x^4-140*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^3*c^3*d*x^4+30*
(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^4*c^4*x^4+20*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c*d^3*x^3-92*(a*c)^
(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c^2*d^2*x^3-932*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^3*d*x^3-20*(
a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^3*c^4*x^3-16*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c^2*d^2*x^2-1024*(
a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c^3*d*x^2-496*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^4*x^2-352
*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*c^3*d*x-672*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*c^4*x-256*((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((b*x+a)^(5/2)*(d*x+c)^(3/2)/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 = 7.28, size = 730, normalized size = 3.05 \begin {gather*} \left [-\frac {15 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - 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 (128 \, a^{5} c^{5} - {\left (15 \, a b^{4} c^{5} - 70 \, a^{2} b^{3} c^{4} d - 128 \, a^{3} b^{2} c^{3} d^{2} + 70 \, a^{4} b c^{2} d^{3} - 15 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (5 \, a^{2} b^{3} c^{5} + 233 \, a^{3} b^{2} c^{4} d + 23 \, a^{4} b c^{3} d^{2} - 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (31 \, a^{3} b^{2} c^{5} + 64 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 16 \, {\left (21 \, a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2560 \, a^{3} c^{4} x^{5}}, \frac {15 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - 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 (128 \, a^{5} c^{5} - {\left (15 \, a b^{4} c^{5} - 70 \, a^{2} b^{3} c^{4} d - 128 \, a^{3} b^{2} c^{3} d^{2} + 70 \, a^{4} b c^{2} d^{3} - 15 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (5 \, a^{2} b^{3} c^{5} + 233 \, a^{3} b^{2} c^{4} d + 23 \, a^{4} b c^{3} d^{2} - 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (31 \, a^{3} b^{2} c^{5} + 64 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 16 \, {\left (21 \, a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1280 \, a^{3} c^{4} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^6,x, algorithm="fricas")

[Out]

[-1/2560*(15*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*sqr
t(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*(128*a^5*c^5 - (15*a*b^4*c^5 - 70*a^2*b^3*c^4*d - 128*
a^3*b^2*c^3*d^2 + 70*a^4*b*c^2*d^3 - 15*a^5*c*d^4)*x^4 + 2*(5*a^2*b^3*c^5 + 233*a^3*b^2*c^4*d + 23*a^4*b*c^3*d
^2 - 5*a^5*c^2*d^3)*x^3 + 8*(31*a^3*b^2*c^5 + 64*a^4*b*c^4*d + a^5*c^3*d^2)*x^2 + 16*(21*a^4*b*c^5 + 11*a^5*c^
4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c^4*x^5), 1/1280*(15*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 -
10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*sqrt(-a*c)*x^5*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sq
rt(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*(128*a^5*c^5 - (15*a*b^4*c^5 -
70*a^2*b^3*c^4*d - 128*a^3*b^2*c^3*d^2 + 70*a^4*b*c^2*d^3 - 15*a^5*c*d^4)*x^4 + 2*(5*a^2*b^3*c^5 + 233*a^3*b^2
*c^4*d + 23*a^4*b*c^3*d^2 - 5*a^5*c^2*d^3)*x^3 + 8*(31*a^3*b^2*c^5 + 64*a^4*b*c^4*d + a^5*c^3*d^2)*x^2 + 16*(2
1*a^4*b*c^5 + 11*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c^4*x^5)]

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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 {3}{2}}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(d*x+c)**(3/2)/x**6,x)

[Out]

Integral((a + b*x)**(5/2)*(c + d*x)**(3/2)/x**6, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 5927 vs. \(2 (195) = 390\).
time = 52.29, size = 5927, normalized size = 24.80 \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)^(3/2)/x^6,x, algorithm="giac")

[Out]

-1/640*(15*(sqrt(b*d)*b^6*c^5*abs(b) - 5*sqrt(b*d)*a*b^5*c^4*d*abs(b) + 10*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) - sqrt(b*d)*a^5*b*d^5*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))/(sq
rt(-a*b*c*d)*a^2*b*c^3) - 2*(15*sqrt(b*d)*b^24*c^14*abs(b) - 220*sqrt(b*d)*a*b^23*c^13*d*abs(b) + 1247*sqrt(b*
d)*a^2*b^22*c^12*d^2*abs(b) - 3600*sqrt(b*d)*a^3*b^21*c^11*d^3*abs(b) + 5075*sqrt(b*d)*a^4*b^20*c^10*d^4*abs(b
) + 180*sqrt(b*d)*a^5*b^19*c^9*d^5*abs(b) - 15165*sqrt(b*d)*a^6*b^18*c^8*d^6*abs(b) + 32256*sqrt(b*d)*a^7*b^17
*c^7*d^7*abs(b) - 38595*sqrt(b*d)*a^8*b^16*c^6*d^8*abs(b) + 30540*sqrt(b*d)*a^9*b^15*c^5*d^9*abs(b) - 16595*sq
rt(b*d)*a^10*b^14*c^4*d^10*abs(b) + 6160*sqrt(b*d)*a^11*b^13*c^3*d^11*abs(b) - 1503*sqrt(b*d)*a^12*b^12*c^2*d^
12*abs(b) + 220*sqrt(b*d)*a^13*b^11*c*d^13*abs(b) - 15*sqrt(b*d)*a^14*b^10*d^14*abs(b) - 135*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) + 1555*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) - 6730*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) + 14490*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) - 16165*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) + 9185*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^17*c^8*d^5*abs(b) - 9340*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) + 27260*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) - 45025*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) + 41765*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^13*c^4*d^9*abs(b) - 23450*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^12*c^3*d^10*abs(b) + 8010*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^11*c^2*d^11*abs(b) - 1555*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) + 135*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) + 540*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) - 4800*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) + 13200*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) - 13760*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) + 5260*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) - 11520*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^1
5*c^7*d^5*abs(b) + 25600*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) - 8960*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) - 25740*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) + 34240*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) - 18320*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) + 4800*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) - 540*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) - 1
260*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) + 8540*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) - 8620*sqrt(b*d)*(s
qrt(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) - 11780*sqrt(b*d)*(sqr
t(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) + 18280*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) - 8840*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) + 13960*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*abs(b) - 2920*sqrt(b*d)*(sqrt(b*d)*sqr
t(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) - 24060*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) + 23980*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) - 8540*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^1...

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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 )}^{3/2}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^6,x)

[Out]

int(((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^6, x)

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