3.6.39 \(\int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^5} \, dx\)
Optimal. Leaf size=233 \[ \frac {(3 a d+5 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{7/2} c^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+5 b c) (b c-a d)^2}{64 a^3 c^2 x}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (3 a d+5 b c) (b c-a d)}{96 a^2 c^2 x^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{24 a c^2 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4} \]
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Rubi [A] time = 0.12, 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 =
{96, 94, 93, 208} \begin {gather*} \frac {\sqrt {a+b x} (c+d x)^{3/2} (3 a d+5 b c) (b c-a d)}{96 a^2 c^2 x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+5 b c) (b c-a d)^2}{64 a^3 c^2 x}+\frac {(3 a d+5 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^{7/2} c^{5/2}}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{24 a c^2 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(Sqrt[a + b*x]*(c + d*x)^(3/2))/x^5,x]
[Out]
-((b*c - a*d)^2*(5*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^3*c^2*x) + ((b*c - a*d)*(5*b*c + 3*a*d)*Sqr
t[a + b*x]*(c + d*x)^(3/2))/(96*a^2*c^2*x^2) + ((5*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*a*c^2*x^3)
- ((a + b*x)^(3/2)*(c + d*x)^(5/2))/(4*a*c*x^4) + ((b*c - a*d)^3*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x
])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/2)*c^(5/2))
Rule 93
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 94
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[p, 1] && !SumSimplerQ[m, 1])
Rule 96
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^5} \, dx &=-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}-\frac {\left (\frac {5 b c}{2}+\frac {3 a d}{2}\right ) \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^4} \, dx}{4 a c}\\ &=\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 a c^2 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}-\frac {((b c-a d) (5 b c+3 a d)) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx}{48 a c^2}\\ &=\frac {(b c-a d) (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 a^2 c^2 x^2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 a c^2 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}+\frac {\left ((b c-a d)^2 (5 b c+3 a d)\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{64 a^2 c^2}\\ &=-\frac {(b c-a d)^2 (5 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a^3 c^2 x}+\frac {(b c-a d) (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 a^2 c^2 x^2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 a c^2 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}-\frac {\left ((b c-a d)^3 (5 b c+3 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a^3 c^2}\\ &=-\frac {(b c-a d)^2 (5 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a^3 c^2 x}+\frac {(b c-a d) (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 a^2 c^2 x^2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 a c^2 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}-\frac {\left ((b c-a d)^3 (5 b c+3 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 a^3 c^2}\\ &=-\frac {(b c-a d)^2 (5 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a^3 c^2 x}+\frac {(b c-a d) (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 a^2 c^2 x^2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 a c^2 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}+\frac {(b c-a d)^3 (5 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{7/2} c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 179, normalized size = 0.77 \begin {gather*} \frac {\frac {x (3 a d+5 b c) \left (\frac {x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt {c}}+8 \sqrt {a+b x} (c+d x)^{5/2}\right )}{c}-48 (a+b x)^{3/2} (c+d x)^{5/2}}{192 a c x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(Sqrt[a + b*x]*(c + d*x)^(3/2))/x^5,x]
[Out]
(-48*(a + b*x)^(3/2)*(c + d*x)^(5/2) + ((5*b*c + 3*a*d)*x*(8*Sqrt[a + b*x]*(c + d*x)^(5/2) + ((b*c - a*d)*x*(S
qrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c - 3*b*c*x + 5*a*d*x) + 3*(b*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*
Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(5/2)*Sqrt[c])))/c)/(192*a*c*x^4)
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IntegrateAlgebraic [A] time = 0.41, size = 260, normalized size = 1.12 \begin {gather*} \frac {\sqrt {a+b x} (a d-b c)^3 \left (9 a^4 d-\frac {33 a^3 c d (a+b x)}{c+d x}+15 a^3 b c+\frac {73 a^2 b c^2 (a+b x)}{c+d x}-\frac {33 a^2 c^2 d (a+b x)^2}{(c+d x)^2}+\frac {15 b c^4 (a+b x)^3}{(c+d x)^3}-\frac {55 a b c^3 (a+b x)^2}{(c+d x)^2}+\frac {9 a c^3 d (a+b x)^3}{(c+d x)^3}\right )}{192 a^3 c^2 \sqrt {c+d x} \left (a-\frac {c (a+b x)}{c+d x}\right )^4}-\frac {(a d-b c)^3 (3 a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{7/2} c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(Sqrt[a + b*x]*(c + d*x)^(3/2))/x^5,x]
[Out]
((-(b*c) + a*d)^3*Sqrt[a + b*x]*(15*a^3*b*c + 9*a^4*d + (15*b*c^4*(a + b*x)^3)/(c + d*x)^3 + (9*a*c^3*d*(a + b
*x)^3)/(c + d*x)^3 - (55*a*b*c^3*(a + b*x)^2)/(c + d*x)^2 - (33*a^2*c^2*d*(a + b*x)^2)/(c + d*x)^2 + (73*a^2*b
*c^2*(a + b*x))/(c + d*x) - (33*a^3*c*d*(a + b*x))/(c + d*x)))/(192*a^3*c^2*Sqrt[c + d*x]*(a - (c*(a + b*x))/(
c + d*x))^4) - ((-(b*c) + a*d)^3*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64
*a^(7/2)*c^(5/2))
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fricas [A] time = 7.78, size = 570, normalized size = 2.45 \begin {gather*} \left [-\frac {3 \, {\left (5 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} - 3 \, 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 (15 \, a b^{3} c^{4} - 31 \, a^{2} b^{2} c^{3} d + 9 \, a^{3} b c^{2} d^{2} - 9 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} - 10 \, a^{3} b c^{3} d - 3 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} + 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{4} c^{3} x^{4}}, -\frac {3 \, {\left (5 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} - 3 \, 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 (15 \, a b^{3} c^{4} - 31 \, a^{2} b^{2} c^{3} d + 9 \, a^{3} b c^{2} d^{2} - 9 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} - 10 \, a^{3} b c^{3} d - 3 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} + 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{4} c^{3} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(3/2)*(b*x+a)^(1/2)/x^5,x, algorithm="fricas")
[Out]
[-1/768*(3*(5*b^4*c^4 - 12*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 3*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 + (15*a*b^3*c^4 - 31*a^2*b^2*c^3*d + 9*a^3*b*c^2*d^2 - 9*a^4*
c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 - 10*a^3*b*c^3*d - 3*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + 9*a^4*c^3*d)*x)*sqrt(b*x
+ a)*sqrt(d*x + c))/(a^4*c^3*x^4), -1/384*(3*(5*b^4*c^4 - 12*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 -
3*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 + (15*a*b^3*c^4 - 31*a^2*b^2*c^3*d + 9*a^3*b*c^2*d^2 -
9*a^4*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 - 10*a^3*b*c^3*d - 3*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + 9*a^4*c^3*d)*x)*sq
rt(b*x + a)*sqrt(d*x + c))/(a^4*c^3*x^4)]
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giac [B] time = 16.75, size = 3834, normalized size = 16.45
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(3/2)*(b*x+a)^(1/2)/x^5,x, algorithm="giac")
[Out]
1/192*(3*(5*sqrt(b*d)*b^5*c^4*abs(b) - 12*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) +
4*sqrt(b*d)*a^3*b^2*c*d^3*abs(b) - 3*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^3*b*c^2) - 2*(15*sqrt
(b*d)*b^19*c^11*abs(b) - 151*sqrt(b*d)*a*b^18*c^10*d*abs(b) + 677*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) - 1789*sqr
t(b*d)*a^3*b^16*c^8*d^3*abs(b) + 3110*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) - 3766*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(
b) + 3290*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b) - 2122*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) + 1019*sqrt(b*d)*a^8*b^11
*c^3*d^8*abs(b) - 355*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) + 81*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) - 9*sqrt(b*d)*a^
11*b^8*d^11*abs(b) - 105*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) + 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*b^16*c^9*d*abs(b)
- 1933*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
328*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) - 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^4*b^13*c^6*d^4*abs(b) - 2068*sqr
t(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) + 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^6*b^11*c^4*d^6*abs(b) - 2152*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) + 1067*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) - 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^9*b^8*c*d^9*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*a^10*b^7*d^10*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*b^15*c^9*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*b^14*c^8*d*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^2*b^13*c^7*d^2*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^3*b^12*c^6*d^3*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^4*b^11*c^5*d^4*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^5*b^10*c^4*d^5*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^6*b^9*c^3*d^6*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^7*b^8*c^2*d^7*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^8*b^7*c*d^8*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*a^9*b^
6*d^9*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*b^13*c^8*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*b^12*c^7*d*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^2*b^11*c^6*d^2*abs(b) + 832*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) + 976*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) + 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^6*b^7*c^2*d^6*abs(b) - 480*sqrt(b*d)*(sqrt(b*d)*s
qrt(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) + 315*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) + 525*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^11*c^7*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*b^10*c^6*d*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^2*b^9*c^5*d^2*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^3*b^8*c^4*d^3*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^4*b^7*c^3*d^4*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^5*b^6*c^2*d^5*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^
6*b^5*c*d^6*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*a^7*b^4*d
^7*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))^10*b^9*c^6*abs(b) +
346*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) + 291*sqrt
(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b^7*c^4*d^2*abs(b) + 12*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^3*b^6*c^3*d^3*abs(b) + 1531*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^4*b^5*c^2*d^4*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^5*b^4*c*d^5*abs(b) + 189*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^6*b^3*d^6*abs(b) + 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^7*c^5*abs(b) - 147*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^
2*c + (b*x + a)*b*d - a*b*d))^12*a*b^6*c^4*d*abs(b) - 126*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^12*a^2*b^5*c^3*d^2*abs(b) - 558*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^12*a^3*b^4*c^2*d^3*abs(b) + 21*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^12*a^4*b^3*c*d^4*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))^12*a^5*b^2*d^5*abs(b) - 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*
b^5*c^4*abs(b) + 36*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a*b^4*c^3*d*a
bs(b) - 18*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^2*b^3*c^2*d^2*abs(b)
- 12*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^3*b^2*c*d^3*abs(b) + 9*sq
rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^4*b*d^4*abs(b))/((b^4*c^2 - 2*a*b
^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b*d
)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^4)^4*a^3*c^2))/b
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maple [B] time = 0.02, size = 705, normalized size = 3.03 \begin {gather*} -\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (9 a^{4} d^{4} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-12 a^{3} b c \,d^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-18 a^{2} b^{2} c^{2} d^{2} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+36 a \,b^{3} c^{3} d \,x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-15 b^{4} c^{4} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-18 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} d^{3} x^{3}+18 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b c \,d^{2} x^{3}-62 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{2} c^{2} d \,x^{3}+30 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{3} c^{3} x^{3}+12 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} c \,d^{2} x^{2}+40 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b \,c^{2} d \,x^{2}-20 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{2} c^{3} x^{2}+144 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} c^{2} d x +16 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b \,c^{3} x +96 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{3} c^{3}\right )}{384 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{3} c^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^(3/2)*(b*x+a)^(1/2)/x^5,x)
[Out]
-1/384*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^3/c^2*(9*a^4*d^4*x^4*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b
*c*x+a*c)^(1/2))/x)-12*a^3*b*c*d^3*x^4*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)
-18*a^2*b^2*c^2*d^2*x^4*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)+36*a*b^3*c^3*d
*x^4*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)-15*b^4*c^4*x^4*ln((a*d*x+b*c*x+2*
a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)-18*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*d^3*x
^3+18*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b*c*d^2*x^3-62*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/
2)*a*b^2*c^2*d*x^3+30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^3*c^3*x^3+12*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*
c*x+a*c)^(1/2)*a^3*c*d^2*x^2+40*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b*c^2*d*x^2-20*(a*c)^(1/2)*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^2*c^3*x^2+144*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*c^2*d*x+16*(a*c
)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b*c^3*x+96*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(a*c)^(1/2)*a^3*c^3)/(b
*d*x^2+a*d*x+b*c*x+a*c)^(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((d*x+c)^(3/2)*(b*x+a)^(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 details)Is a*d-b*c zero or nonzero?
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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 )}^{3/2}}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((a + b*x)^(1/2)*(c + d*x)^(3/2))/x^5,x)
[Out]
int(((a + b*x)^(1/2)*(c + d*x)^(3/2))/x^5, x)
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sympy [F(-1)] 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)**(3/2)*(b*x+a)**(1/2)/x**5,x)
[Out]
Timed out
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