3.7.96 \(\int \frac {(c+d x)^{5/2}}{x^5 \sqrt {a+b x}} \, dx\)
Optimal. Leaf size=229 \[ -\frac {5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{9/2} c^{3/2}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2 (a d+7 b c)}{64 a^4 c x}-\frac {5 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d) (a d+7 b c)}{96 a^3 c x^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (a d+7 b c)}{24 a^2 c x^3}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 a c x^4} \]
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Rubi [A] time = 0.12, 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} \begin {gather*} -\frac {5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{9/2} c^{3/2}}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (a d+7 b c)}{24 a^2 c x^3}-\frac {5 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d) (a d+7 b c)}{96 a^3 c x^2}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2 (a d+7 b c)}{64 a^4 c x}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 a c x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^(5/2)/(x^5*Sqrt[a + b*x]),x]
[Out]
(5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^4*c*x) - (5*(b*c - a*d)*(7*b*c + a*d)*Sqrt[a
+ b*x]*(c + d*x)^(3/2))/(96*a^3*c*x^2) + ((7*b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*a^2*c*x^3) - (Sqrt
[a + b*x]*(c + d*x)^(7/2))/(4*a*c*x^4) - (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[
a]*Sqrt[c + d*x])])/(64*a^(9/2)*c^(3/2))
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 {(c+d x)^{5/2}}{x^5 \sqrt {a+b x}} \, dx &=-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 a c x^4}-\frac {\left (\frac {7 b c}{2}+\frac {a d}{2}\right ) \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx}{4 a c}\\ &=\frac {(7 b c+a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 a^2 c x^3}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 a c x^4}+\frac {(5 (b c-a d) (7 b c+a d)) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx}{48 a^2 c}\\ &=-\frac {5 (b c-a d) (7 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 a^3 c x^2}+\frac {(7 b c+a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 a^2 c x^3}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 a c x^4}-\frac {\left (5 (b c-a d)^2 (7 b c+a d)\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{64 a^3 c}\\ &=\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a^4 c x}-\frac {5 (b c-a d) (7 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 a^3 c x^2}+\frac {(7 b c+a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 a^2 c x^3}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 a c x^4}+\frac {\left (5 (b c-a d)^3 (7 b c+a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a^4 c}\\ &=\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a^4 c x}-\frac {5 (b c-a d) (7 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 a^3 c x^2}+\frac {(7 b c+a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 a^2 c x^3}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 a c x^4}+\frac {\left (5 (b c-a d)^3 (7 b c+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^4 c}\\ &=\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a^4 c x}-\frac {5 (b c-a d) (7 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 a^3 c x^2}+\frac {(7 b c+a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 a^2 c x^3}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 a c x^4}-\frac {5 (b c-a d)^3 (7 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{9/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 183, normalized size = 0.80 \begin {gather*} -\frac {\frac {(a d+7 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+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 )}{48 a x^3}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{x^4}}{4 a c} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^(5/2)/(x^5*Sqrt[a + b*x]),x]
[Out]
-1/4*((Sqrt[a + b*x]*(c + d*x)^(7/2))/x^4 + ((7*b*c + a*d)*(-8*Sqrt[a + b*x]*(c + d*x)^(5/2) + (5*(b*c - a*d)*
x*(Sqrt[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])))/(48*a*x^3))/(a*c)
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IntegrateAlgebraic [A] time = 0.38, size = 259, normalized size = 1.13 \begin {gather*} \frac {5 (a d-b c)^3 (a d+7 b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{9/2} c^{3/2}}-\frac {\sqrt {a+b x} (a d-b c)^3 \left (15 a^4 d+\frac {73 a^3 c d (a+b x)}{c+d x}-279 a^3 b c+\frac {511 a^2 b c^2 (a+b x)}{c+d x}-\frac {55 a^2 c^2 d (a+b x)^2}{(c+d x)^2}+\frac {105 b c^4 (a+b x)^3}{(c+d x)^3}-\frac {385 a b c^3 (a+b x)^2}{(c+d x)^2}+\frac {15 a c^3 d (a+b x)^3}{(c+d x)^3}\right )}{192 a^4 c \sqrt {c+d x} \left (a-\frac {c (a+b x)}{c+d x}\right )^4} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(c + d*x)^(5/2)/(x^5*Sqrt[a + b*x]),x]
[Out]
-1/192*((-(b*c) + a*d)^3*Sqrt[a + b*x]*(-279*a^3*b*c + 15*a^4*d + (105*b*c^4*(a + b*x)^3)/(c + d*x)^3 + (15*a*
c^3*d*(a + b*x)^3)/(c + d*x)^3 - (385*a*b*c^3*(a + b*x)^2)/(c + d*x)^2 - (55*a^2*c^2*d*(a + b*x)^2)/(c + d*x)^
2 + (511*a^2*b*c^2*(a + b*x))/(c + d*x) + (73*a^3*c*d*(a + b*x))/(c + d*x)))/(a^4*c*Sqrt[c + d*x]*(a - (c*(a +
b*x))/(c + d*x))^4) + (5*(-(b*c) + a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x
])])/(64*a^(9/2)*c^(3/2))
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fricas [A] time = 6.59, size = 574, normalized size = 2.51 \begin {gather*} \left [-\frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{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 (105 \, a b^{3} c^{4} - 265 \, a^{2} b^{2} c^{3} d + 191 \, a^{3} b c^{2} d^{2} - 15 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (35 \, a^{2} b^{2} c^{4} - 86 \, a^{3} b c^{3} d + 59 \, a^{4} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (7 \, a^{3} b c^{4} - 17 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{5} c^{2} x^{4}}, \frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{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 (105 \, a b^{3} c^{4} - 265 \, a^{2} b^{2} c^{3} d + 191 \, a^{3} b c^{2} d^{2} - 15 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (35 \, a^{2} b^{2} c^{4} - 86 \, a^{3} b c^{3} d + 59 \, a^{4} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (7 \, a^{3} b c^{4} - 17 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{5} c^{2} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(5/2)/x^5/(b*x+a)^(1/2),x, algorithm="fricas")
[Out]
[-1/768*(15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^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 - (105*a*b^3*c^4 - 265*a^2*b^2*c^3*d + 191*a^3*b*c^2*d^2 - 15
*a^4*c*d^3)*x^3 + 2*(35*a^2*b^2*c^4 - 86*a^3*b*c^3*d + 59*a^4*c^2*d^2)*x^2 - 8*(7*a^3*b*c^4 - 17*a^4*c^3*d)*x)
*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^2*x^4), 1/384*(15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^
3*b*c*d^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 - (105*a*b^3*c^4 - 265*a^2*b^2*c^3*d + 191*a^
3*b*c^2*d^2 - 15*a^4*c*d^3)*x^3 + 2*(35*a^2*b^2*c^4 - 86*a^3*b*c^3*d + 59*a^4*c^2*d^2)*x^2 - 8*(7*a^3*b*c^4 -
17*a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^2*x^4)]
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giac [B] time = 12.10, size = 3834, normalized size = 16.74
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(5/2)/x^5/(b*x+a)^(1/2),x, algorithm="giac")
[Out]
-1/192*(15*(7*sqrt(b*d)*b^5*c^4*abs(b) - 20*sqrt(b*d)*a*b^4*c^3*d*abs(b) + 18*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) - 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^4*b*c) - 2*(105*sqrt
(b*d)*b^19*c^11*abs(b) - 1105*sqrt(b*d)*a*b^18*c^10*d*abs(b) + 5251*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) - 14843*
sqrt(b*d)*a^3*b^16*c^8*d^3*abs(b) + 27658*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) - 35546*sqrt(b*d)*a^5*b^14*c^6*d^5
*abs(b) + 31990*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b) - 20006*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) + 8413*sqrt(b*d)*a
^8*b^11*c^3*d^8*abs(b) - 2213*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) + 311*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) - 15*sq
rt(b*d)*a^11*b^8*d^11*abs(b) - 735*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) + 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*b^16*c^
9*d*abs(b) - 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^2*b^15*c^8*d^
2*abs(b) + 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^3*b^14*c^7*d^3*
abs(b) - 8782*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) - 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*abs(b
) + 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^6*b^11*c^4*d^6*abs(b)
- 28312*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) +
11245*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*abs(b) - 209
8*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)*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) + 2205*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) - 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*b^14*c^8*d*abs(b) + 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^2*b^13*c^7*d^2*abs(b) - 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^3*b^12*c^6*d^3*abs(b) + 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^4*b^11*c^5*d^4*abs(b) - 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^5*b^10*c^4*d^5*abs(b) + 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^6*b^9*c^3*d^6*abs(b) - 21580*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^7*b^8*c^2*d^7*abs(b) + 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^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) - 3675*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) + 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*b^12*c^7*d*abs(b) - 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^2*b^11*c^6*d^2*abs(b) + 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
^3*b^10*c^5*d^3*abs(b) + 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*abs(b) - 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^5*b^8
*c^3*d^5*abs(b) + 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^6*b^7*c^
2*d^6*abs(b) - 8240*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) +
3675*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) - 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*b^10*c^6*d*abs(b) + 3275*sqrt(b*d)*(sq
rt(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) + 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^3*b^8*c^4*d^3*abs(b) + 1513*sqrt(b*d)*(sqrt(b*d)*s
qrt(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) - 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^5*b^6*c^2*d^5*abs(b) + 6385*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) - 2205*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) + 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*b^8*c^5*d*abs(b) + 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^2*b^7*c^4*d^2*abs(b) + 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*abs(b) - 1667*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*abs(b) - 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^5*b^4*c*d^5*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*a^6
*b^3*d^6*abs(b) + 735*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) - 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*b^6*c^4*d*abs(b) - 2
10*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) + 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^3*b^4*c^2*d^3*abs(b) + 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^4*b^3*c*d^4*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*a^5*b^2*d^5*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))^14*b^5*c^4*abs(b) + 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*b^4*c^3*d*abs(b) - 270*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^2*b^3*c^2*d^2*abs(b) + 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^3*b^2*c*d^3*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*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^4*c))/b
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maple [B] time = 0.03, size = 593, normalized size = 2.59 \begin {gather*} \frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (15 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 )+60 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 )+300 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 )-105 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 )-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} d^{3} x^{3}+382 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b c \,d^{2} x^{3}-530 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{3}+210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{3} c^{3} x^{3}-236 \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}-140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-272 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{2} d x +112 \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^{4} c \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^(5/2)/x^5/(b*x+a)^(1/2),x)
[Out]
1/384*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^4/c*(15*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)+60*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)+300*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)-105*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)-30*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*d^3*x^3+382*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*
b*c*d^2*x^3-530*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^2*c^2*d*x^3+210*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*b^
3*c^3*x^3-236*((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^2-140*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^2*c^3*x^2-272*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*c^
2*d*x+112*((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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(5/2)/x^5/(b*x+a)^(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 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{x^5\,\sqrt {a+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)^(5/2)/(x^5*(a + b*x)^(1/2)),x)
[Out]
int((c + d*x)^(5/2)/(x^5*(a + b*x)^(1/2)), 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)**(5/2)/x**5/(b*x+a)**(1/2),x)
[Out]
Timed out
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