3.7.5 \(\int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^5} \, dx\) [605]

Optimal. Leaf size=233 \[ \frac {(b c-a d)^2 (3 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a^2 c^3 x}+\frac {(b c-a d) (3 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 a c^3 x^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 a c^2 x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 a c x^4}-\frac {(b c-a d)^3 (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{7/2}} \]

[Out]

1/24*(5*a*d+3*b*c)*(b*x+a)^(3/2)*(d*x+c)^(3/2)/a/c^2/x^3-1/4*(b*x+a)^(5/2)*(d*x+c)^(3/2)/a/c/x^4-1/64*(-a*d+b*
c)^3*(5*a*d+3*b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(5/2)/c^(7/2)+1/32*(-a*d+b*c)*(5*a*d
+3*b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a/c^3/x^2+1/64*(-a*d+b*c)^2*(5*a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/
c^3/x

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Rubi [A]
time = 0.08, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {(5 a d+3 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{7/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (5 a d+3 b c) (b c-a d)^2}{64 a^2 c^3 x}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+3 b c) (b c-a d)}{32 a c^3 x^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 a c^2 x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 a c x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^2*(3*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^2*c^3*x) + ((b*c - a*d)*(3*b*c + 5*a*d)*Sqrt
[a + b*x]*(c + d*x)^(3/2))/(32*a*c^3*x^2) + ((3*b*c + 5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(24*a*c^2*x^3) -
((a + b*x)^(5/2)*(c + d*x)^(3/2))/(4*a*c*x^4) - ((b*c - a*d)^3*(3*b*c + 5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x]
)/(Sqrt[a]*Sqrt[c + d*x])])/(64*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 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 {(a+b x)^{3/2} \sqrt {c+d x}}{x^5} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 a c x^4}-\frac {\left (\frac {3 b c}{2}+\frac {5 a d}{2}\right ) \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^4} \, dx}{4 a c}\\ &=\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 a c^2 x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 a c x^4}-\frac {((b c-a d) (3 b c+5 a d)) \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3} \, dx}{16 a c^2}\\ &=\frac {(b c-a d) (3 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 a c^3 x^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 a c^2 x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 a c x^4}-\frac {\left ((b c-a d)^2 (3 b c+5 a d)\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{64 a c^3}\\ &=\frac {(b c-a d)^2 (3 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a^2 c^3 x}+\frac {(b c-a d) (3 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 a c^3 x^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 a c^2 x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 a c x^4}+\frac {\left ((b c-a d)^3 (3 b c+5 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a^2 c^3}\\ &=\frac {(b c-a d)^2 (3 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a^2 c^3 x}+\frac {(b c-a d) (3 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 a c^3 x^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 a c^2 x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 a c x^4}+\frac {\left ((b c-a d)^3 (3 b c+5 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 )}{64 a^2 c^3}\\ &=\frac {(b c-a d)^2 (3 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a^2 c^3 x}+\frac {(b c-a d) (3 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 a c^3 x^2}+\frac {(3 b c+5 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 a c^2 x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 a c x^4}-\frac {(b c-a d)^3 (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 200, normalized size = 0.86 \begin {gather*} \frac {(-b c+a d)^3 \left (\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} \left (-9 b^3 c^3 x^3+3 a b^2 c^2 x^2 (2 c+3 d x)+a^2 b c x \left (72 c^2+20 c d x-31 d^2 x^2\right )+a^3 \left (48 c^3+8 c^2 d x-10 c d^2 x^2+15 d^3 x^3\right )\right )}{(b c-a d)^3 x^4}+3 (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )}{192 a^{5/2} c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-(b*c) + a*d)^3*((Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-9*b^3*c^3*x^3 + 3*a*b^2*c^2*x^2*(2*c + 3*d*x
) + a^2*b*c*x*(72*c^2 + 20*c*d*x - 31*d^2*x^2) + a^3*(48*c^3 + 8*c^2*d*x - 10*c*d^2*x^2 + 15*d^3*x^3)))/((b*c
- a*d)^3*x^4) + 3*(3*b*c + 5*a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])]))/(192*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. \(592\) vs. \(2(195)=390\).
time = 0.07, size = 593, normalized size = 2.55

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4}-36 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4}+18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4}+12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4}-9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4}-30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} d^{3} x^{3}+62 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b c \,d^{2} x^{3}-18 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c^{2} d \,x^{3}+18 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{3} c^{3} x^{3}+20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c \,d^{2} x^{2}-40 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b \,c^{2} d \,x^{2}-12 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c^{3} x^{2}-16 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c^{2} d x -144 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b \,c^{3} x -96 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c^{3} \sqrt {a c}\right )}{384 a^{2} c^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{4} \sqrt {a c}}\)	\(593\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

1/384*(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^4*d^4*x^4-36*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*c*d^3*x^4+18*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^2*c^2*d^2*x^4+12*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^3*c^3*d*x^4-9*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^4*c^4*x^4-30*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*d^3*x^3+62*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b*c
*d^2*x^3-18*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^2*c^2*d*x^3+18*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^3*c^3
*x^3+20*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*c*d^2*x^2-40*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b*c^2*d*x
^2-12*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^2*c^3*x^2-16*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*c^2*d*x-144
*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b*c^3*x-96*((d*x+c)*(b*x+a))^(1/2)*a^3*c^3*(a*c)^(1/2))/((d*x+c)*(b*x
+a))^(1/2)/x^4/(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)^(3/2)*(d*x+c)^(1/2)/x^5,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 = 4.12, size = 572, normalized size = 2.45 \begin {gather*} \left [-\frac {3 \, {\left (3 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \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 (48 \, a^{4} c^{4} - {\left (9 \, a b^{3} c^{4} - 9 \, a^{2} b^{2} c^{3} d + 31 \, a^{3} b c^{2} d^{2} - 15 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{2} c^{4} + 10 \, a^{3} b c^{3} d - 5 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (9 \, a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{3} c^{4} x^{4}}, \frac {3 \, {\left (3 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \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 (48 \, a^{4} c^{4} - {\left (9 \, a b^{3} c^{4} - 9 \, a^{2} b^{2} c^{3} d + 31 \, a^{3} b c^{2} d^{2} - 15 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{2} c^{4} + 10 \, a^{3} b c^{3} d - 5 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (9 \, a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{3} c^{4} x^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(3*(3*b^4*c^4 - 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 - 5*a^4*d^4)*sqrt(a*c)*x^4*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*(48*a^4*c^4 - (9*a*b^3*c^4 - 9*a^2*b^2*c^3*d + 31*a^3*b*c^2*d^2 - 15*a^4*
c*d^3)*x^3 + 2*(3*a^2*b^2*c^4 + 10*a^3*b*c^3*d - 5*a^4*c^2*d^2)*x^2 + 8*(9*a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(b*x
+ a)*sqrt(d*x + c))/(a^3*c^4*x^4), 1/384*(3*(3*b^4*c^4 - 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 -
5*a^4*d^4)*sqrt(-a*c)*x^4*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*(48*a^4*c^4 - (9*a*b^3*c^4 - 9*a^2*b^2*c^3*d + 31*a^3*b*c^2*d^2 - 1
5*a^4*c*d^3)*x^3 + 2*(3*a^2*b^2*c^4 + 10*a^3*b*c^3*d - 5*a^4*c^2*d^2)*x^2 + 8*(9*a^3*b*c^4 + a^4*c^3*d)*x)*sqr
t(b*x + a)*sqrt(d*x + c))/(a^3*c^4*x^4)]

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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 {3}{2}} \sqrt {c + d x}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3834 vs. \(2 (195) = 390\).
time = 3.02, size = 3834, normalized size = 16.45 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(3*(3*sqrt(b*d)*b^5*c^4*abs(b) - 4*sqrt(b*d)*a*b^4*c^3*d*abs(b) - 6*sqrt(b*d)*a^2*b^3*c^2*d^2*abs(b) +
12*sqrt(b*d)*a^3*b^2*c*d^3*abs(b) - 5*sqrt(b*d)*a^4*b*d^4*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^2*b*c^3) - 2*(9*sqrt
(b*d)*b^19*c^11*abs(b) - 81*sqrt(b*d)*a*b^18*c^10*d*abs(b) + 355*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) - 1019*sqrt
(b*d)*a^3*b^16*c^8*d^3*abs(b) + 2122*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) - 3290*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b
) + 3766*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b) - 3110*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) + 1789*sqrt(b*d)*a^8*b^11*
c^3*d^8*abs(b) - 677*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) + 151*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) - 15*sqrt(b*d)*a
^11*b^8*d^11*abs(b) - 63*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^17*c^10
*abs(b) + 366*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^16*c^9*d*abs(b)
- 1067*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^15*c^8*d^2*abs(b) + 2
152*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^14*c^7*d^3*abs(b) - 2958
*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^13*c^6*d^4*abs(b) + 2068*sq
rt(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^12*c^5*d^5*abs(b) + 514*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^11*c^4*d^6*abs(b) - 2328*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^10*c^3*d^7*abs(b) + 1933*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^8*b^9*c^2*d^8*abs(b) - 722*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^8*c*d^9*abs(b) + 105*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^7*d^10*abs(b) + 189*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^15*c^9*abs(b) - 627*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^14*c^8*d*abs(b) + 820*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^13*c^7*d^2*abs(b) - 348*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^12*c^6*d^3*abs(b) - 762*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^11*c^5*d^4*abs(b) + 1174*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^10*c^4*d^5*abs(b) + 420*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^9*c^3*d^6*abs(b) - 1932*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^8*c^2*d^7*abs(b) + 1381*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^7*c*d^8*abs(b) - 315*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
^6*d^9*abs(b) - 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^13*c^8*abs(b
) + 480*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^12*c^7*d*abs(b) - 220*
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^11*c^6*d^2*abs(b) - 976*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^10*c^5*d^3*abs(b) + 1910*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^9*c^4*d^4*abs(b) - 832*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^5*b^8*c^3*d^5*abs(b) + 788*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^7*c^2*d^6*abs(b) - 1360*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^6*c*d^7*abs(b) + 525*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^5*d^8*abs(b) + 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^11*c^7*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))^8*a*b^10*c^6*d*abs(b) + 1067*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^8*a^2*b^9*c^5*d^2*abs(b) + 113*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^8*a^3*b^8*c^4*d^3*abs(b) + 393*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^8*a^4*b^7*c^3*d^4*abs(b) + 35*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^8*a^5*b^6*c^2*d^5*abs(b) + 785*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a
^6*b^5*c*d^6*abs(b) - 525*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^7*b^4*
d^7*abs(b) - 189*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^9*c^6*abs(b) +
6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^8*c^5*d*abs(b) - 1531*sqrt
(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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