3.681 \(\int \frac {(a+b x)^{5/2}}{x^5 \sqrt {c+d x}} \, dx\)
Optimal. Leaf size=229 \[ \frac {5 (b c-a d)^3 (7 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{9/2}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2 (7 a d+b c)}{64 a c^4 x}+\frac {5 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d) (7 a d+b c)}{96 a c^3 x^2}+\frac {(a+b x)^{5/2} \sqrt {c+d x} (7 a d+b c)}{24 a c^2 x^3}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4} \]
[Out]
5/64*(-a*d+b*c)^3*(7*a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(3/2)/c^(9/2)+5/96*(-a*d+
b*c)*(7*a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/a/c^3/x^2+1/24*(7*a*d+b*c)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/a/c^2/x^3-
1/4*(b*x+a)^(7/2)*(d*x+c)^(1/2)/a/c/x^4+5/64*(-a*d+b*c)^2*(7*a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^4/x
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Rubi [A] time = 0.11, antiderivative size = 229, 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} \[ \frac {5 (b c-a d)^3 (7 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{9/2}}+\frac {(a+b x)^{5/2} \sqrt {c+d x} (7 a d+b c)}{24 a c^2 x^3}+\frac {5 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d) (7 a d+b c)}{96 a c^3 x^2}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2 (7 a d+b c)}{64 a c^4 x}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^(5/2)/(x^5*Sqrt[c + d*x]),x]
[Out]
(5*(b*c - a*d)^2*(b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a*c^4*x) + (5*(b*c - a*d)*(b*c + 7*a*d)*(a + b
*x)^(3/2)*Sqrt[c + d*x])/(96*a*c^3*x^2) + ((b*c + 7*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*a*c^2*x^3) - ((a +
b*x)^(7/2)*Sqrt[c + d*x])/(4*a*c*x^4) + (5*(b*c - a*d)^3*(b*c + 7*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[
a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(9/2))
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 {(a+b x)^{5/2}}{x^5 \sqrt {c+d x}} \, dx &=-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}-\frac {\left (\frac {b c}{2}+\frac {7 a d}{2}\right ) \int \frac {(a+b x)^{5/2}}{x^4 \sqrt {c+d x}} \, dx}{4 a c}\\ &=\frac {(b c+7 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 a c^2 x^3}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}-\frac {(5 (b c-a d) (b c+7 a d)) \int \frac {(a+b x)^{3/2}}{x^3 \sqrt {c+d x}} \, dx}{48 a c^2}\\ &=\frac {5 (b c-a d) (b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{96 a c^3 x^2}+\frac {(b c+7 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 a c^2 x^3}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}-\frac {\left (5 (b c-a d)^2 (b c+7 a d)\right ) \int \frac {\sqrt {a+b x}}{x^2 \sqrt {c+d x}} \, dx}{64 a c^3}\\ &=\frac {5 (b c-a d)^2 (b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^4 x}+\frac {5 (b c-a d) (b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{96 a c^3 x^2}+\frac {(b c+7 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 a c^2 x^3}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}-\frac {\left (5 (b c-a d)^3 (b c+7 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a c^4}\\ &=\frac {5 (b c-a d)^2 (b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^4 x}+\frac {5 (b c-a d) (b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{96 a c^3 x^2}+\frac {(b c+7 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 a c^2 x^3}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}-\frac {\left (5 (b c-a d)^3 (b c+7 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 c^4}\\ &=\frac {5 (b c-a d)^2 (b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^4 x}+\frac {5 (b c-a d) (b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{96 a c^3 x^2}+\frac {(b c+7 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 a c^2 x^3}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}+\frac {5 (b c-a d)^3 (b c+7 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 179, normalized size = 0.78 \[ \frac {\frac {x (7 a d+b c) \left (\frac {5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c-3 a d x+5 b c x)\right )}{\sqrt {a} c^{5/2}}+8 (a+b x)^{5/2} \sqrt {c+d x}\right )}{c}-48 (a+b x)^{7/2} \sqrt {c+d x}}{192 a c x^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^(5/2)/(x^5*Sqrt[c + d*x]),x]
[Out]
(-48*(a + b*x)^(7/2)*Sqrt[c + d*x] + ((b*c + 7*a*d)*x*(8*(a + b*x)^(5/2)*Sqrt[c + d*x] + (5*(b*c - a*d)*x*(Sqr
t[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c + 5*b*c*x - 3*a*d*x) + 3*(b*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sq
rt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(Sqrt[a]*c^(5/2))))/c)/(192*a*c*x^4)
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fricas [A] time = 9.08, size = 570, normalized size = 2.49 \[ \left [-\frac {15 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} - 191 \, a^{2} b^{2} c^{3} d + 265 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{4} - 86 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{2} c^{5} x^{4}}, -\frac {15 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} - 191 \, a^{2} b^{2} c^{3} d + 265 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{4} - 86 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{2} c^{5} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(1/2),x, algorithm="fricas")
[Out]
[-1/768*(15*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*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 - 191*a^2*b^2*c^3*d + 265*a^3*b*c^2*d^2 - 105
*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 - 86*a^3*b*c^3*d + 35*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4 - 7*a^4*c^3*d)*x)
*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^5*x^4), -1/384*(15*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3
*b*c*d^3 - 7*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 - 191*a^2*b^2*c^3*d + 265*a^
3*b*c^2*d^2 - 105*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 - 86*a^3*b*c^3*d + 35*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4
- 7*a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^5*x^4)]
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giac [B] time = 11.22, size = 3689, normalized size = 16.11 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(1/2),x, algorithm="giac")
[Out]
1/192*b*(15*(sqrt(b*d)*b^5*c^4 + 4*sqrt(b*d)*a*b^4*c^3*d - 18*sqrt(b*d)*a^2*b^3*c^2*d^2 + 20*sqrt(b*d)*a^3*b^2
*c*d^3 - 7*sqrt(b*d)*a^4*b*d^4)*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*b*c^4) - 2*(15*sqrt(b*d)*b^19*c^11 - 311*sqrt(b*d)*a*b
^18*c^10*d + 2213*sqrt(b*d)*a^2*b^17*c^9*d^2 - 8413*sqrt(b*d)*a^3*b^16*c^8*d^3 + 20006*sqrt(b*d)*a^4*b^15*c^7*
d^4 - 31990*sqrt(b*d)*a^5*b^14*c^6*d^5 + 35546*sqrt(b*d)*a^6*b^13*c^5*d^6 - 27658*sqrt(b*d)*a^7*b^12*c^4*d^7 +
14843*sqrt(b*d)*a^8*b^11*c^3*d^8 - 5251*sqrt(b*d)*a^9*b^10*c^2*d^9 + 1105*sqrt(b*d)*a^10*b^9*c*d^10 - 105*sqr
t(b*d)*a^11*b^8*d^11 - 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 + 2098*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 - 11245*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^2*b^15*c^8*d^2 + 28312*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^3*b^14*c^7*d^3 - 37250*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 + 20780*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^12*c^5*d^5 + 8782*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^11*c^4*d^6 - 22760*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 + 16043*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^9*c^2*d^8 - 5390*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 + 735*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 + 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 - 5765*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 + 21580*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 - 32868*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 + 21802*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 - 8198*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 + 13212*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 - 18548*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 + 10675*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 - 2205*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
- 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 + 8240*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*b^12*c^7*d - 18740*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 + 14752*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 - 4118*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 - 2960*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^8*c^3*d^5 + 10876*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 - 11200*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 + 3675*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
+ 525*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 - 6385*sqrt(b*d)*(s
qrt(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 + 5725*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 - 1513*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 - 625*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 - 3275*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 + 7175*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 - 3675*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 - 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 + 2426*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 + 1667*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b^7*c^4*d^2 - 180*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 - 325*sqrt(b*d)*(sqrt(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 - 3430*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 + 2205*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^10*a^6*b^3*d^6 + 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 - 243*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 - 1470*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 + 210*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 + 1365*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 - 735*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^5*b^2*d^5 - 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 - 60*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 + 270*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 - 300*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 + 105*sqr
t(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)/((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*c^4))/abs(b)
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maple [B] time = 0.03, size = 593, normalized size = 2.59 \[ -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 a^{4} d^{4} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-300 a^{3} b c \,d^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+270 a^{2} b^{2} c^{2} d^{2} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-60 a \,b^{3} c^{3} d \,x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-15 b^{4} c^{4} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} d^{3} x^{3}+530 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b c \,d^{2} x^{3}-382 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{3}+30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{3} c^{3} x^{3}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-344 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}+236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-112 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{2} d x +272 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{3} x +96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{3}\right )}{384 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,c^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^(5/2)/x^5/(d*x+c)^(1/2),x)
[Out]
-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^4*(105*a^4*d^4*x^4*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c)
)^(1/2))/x)-300*a^3*b*c*d^3*x^4*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)+270*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*x+a)*(d*x+c))^(1/2))/x)-60*a*b^3*c^3*d*x^4*ln((a*d*x+b*c*x+2
*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+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*x+a)*(d*
x+c))^(1/2))/x)-210*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*d^3*x^3+530*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^
2*b*c*d^2*x^3-382*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^2*c^2*d*x^3+30*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*b
^3*c^3*x^3+140*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*c*d^2*x^2-344*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b
*c^2*d*x^2+236*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^2*c^3*x^2-112*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*c
^2*d*x+272*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b*c^3*x+96*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*c^3)/((b
*x+a)*(d*x+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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(1/2),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 \[ \int \frac {{\left (a+b\,x\right )}^{5/2}}{x^5\,\sqrt {c+d\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)^(5/2)/(x^5*(c + d*x)^(1/2)),x)
[Out]
int((a + b*x)^(5/2)/(x^5*(c + d*x)^(1/2)), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**(5/2)/x**5/(d*x+c)**(1/2),x)
[Out]
Timed out
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