3.574 \(\int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=283 \[ -\frac {(3 a d+7 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+7 b c) (b c-a d)^3}{128 a^4 c^2 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (3 a d+7 b c) (b c-a d)^2}{192 a^3 c^2 x^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (3 a d+7 b c) (b c-a d)}{240 a^2 c^2 x^3}+\frac {\sqrt {a+b x} (c+d x)^{7/2} (3 a d+7 b c)}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5} \]

[Out]

-1/5*(b*x+a)^(3/2)*(d*x+c)^(7/2)/a/c/x^5-1/128*(-a*d+b*c)^4*(3*a*d+7*b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2
)/(d*x+c)^(1/2))/a^(9/2)/c^(5/2)-1/192*(-a*d+b*c)^2*(3*a*d+7*b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a^3/c^2/x^2+1/24
0*(-a*d+b*c)*(3*a*d+7*b*c)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/a^2/c^2/x^3+1/40*(3*a*d+7*b*c)*(d*x+c)^(7/2)*(b*x+a)^(1
/2)/a/c^2/x^4+1/128*(-a*d+b*c)^3*(3*a*d+7*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^2/x

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Rubi [A]  time = 0.16, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \[ -\frac {\sqrt {a+b x} (c+d x)^{3/2} (3 a d+7 b c) (b c-a d)^2}{192 a^3 c^2 x^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (3 a d+7 b c) (b c-a d)}{240 a^2 c^2 x^3}+\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+7 b c) (b c-a d)^3}{128 a^4 c^2 x}-\frac {(3 a d+7 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{5/2}}+\frac {\sqrt {a+b x} (c+d x)^{7/2} (3 a d+7 b c)}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^3*(7*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^4*c^2*x) - ((b*c - a*d)^2*(7*b*c + 3*a*d)*S
qrt[a + b*x]*(c + d*x)^(3/2))/(192*a^3*c^2*x^2) + ((b*c - a*d)*(7*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/
(240*a^2*c^2*x^3) + ((7*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/(40*a*c^2*x^4) - ((a + b*x)^(3/2)*(c + d*x
)^(7/2))/(5*a*c*x^5) - ((b*c - a*d)^4*(7*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]
)/(128*a^(9/2)*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)^{5/2}}{x^6} \, dx &=-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac {\left (\frac {7 b c}{2}+\frac {3 a d}{2}\right ) \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^5} \, dx}{5 a c}\\ &=\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac {((b c-a d) (7 b c+3 a d)) \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx}{80 a c^2}\\ &=\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}+\frac {\left ((b c-a d)^2 (7 b c+3 a d)\right ) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx}{96 a^2 c^2}\\ &=-\frac {(b c-a d)^2 (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac {\left ((b c-a d)^3 (7 b c+3 a d)\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{128 a^3 c^2}\\ &=\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^4 c^2 x}-\frac {(b c-a d)^2 (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}+\frac {\left ((b c-a d)^4 (7 b c+3 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^4 c^2}\\ &=\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^4 c^2 x}-\frac {(b c-a d)^2 (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}+\frac {\left ((b c-a d)^4 (7 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 )}{128 a^4 c^2}\\ &=\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^4 c^2 x}-\frac {(b c-a d)^2 (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac {(b c-a d)^4 (7 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.47, size = 218, normalized size = 0.77 \[ -\frac {\frac {(3 a d+7 b c) \left (\frac {x (b c-a d) \left (\frac {5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt {c}}-8 \sqrt {a+b x} (c+d x)^{5/2}\right )}{a}-48 \sqrt {a+b x} (c+d x)^{7/2}\right )}{384 c x^4}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{x^5}}{5 a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(((a + b*x)^(3/2)*(c + d*x)^(7/2))/x^5 + ((7*b*c + 3*a*d)*(-48*Sqrt[a + b*x]*(c + d*x)^(7/2) + ((b*c - a*
d)*x*(-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])))/a))/(384*c*x^4))/(a*c)

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fricas [A]  time = 10.37, size = 732, normalized size = 2.59 \[ \left [\frac {15 \, {\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, 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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 340 \, a^{2} b^{3} c^{4} d + 346 \, a^{3} b^{2} c^{3} d^{2} - 60 \, a^{4} b c^{2} d^{3} + 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 111 \, a^{3} b^{2} c^{4} d + 109 \, a^{4} b c^{3} d^{2} + 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 22 \, a^{4} b c^{4} d - 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + 21 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{5} c^{3} x^{5}}, \frac {15 \, {\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, 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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 340 \, a^{2} b^{3} c^{4} d + 346 \, a^{3} b^{2} c^{3} d^{2} - 60 \, a^{4} b c^{2} d^{3} + 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 111 \, a^{3} b^{2} c^{4} d + 109 \, a^{4} b c^{3} d^{2} + 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 22 \, a^{4} b c^{4} d - 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + 21 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{5} c^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^6,x, algorithm="fricas")

[Out]

[1/7680*(15*(7*b^5*c^5 - 25*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 3*a^5*d^5)
*sqrt(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*(384*a^5*c^5 - (105*a*b^4*c^5 - 340*a^2*b^3*c^4*d
+ 346*a^3*b^2*c^3*d^2 - 60*a^4*b*c^2*d^3 + 45*a^5*c*d^4)*x^4 + 2*(35*a^2*b^3*c^5 - 111*a^3*b^2*c^4*d + 109*a^4
*b*c^3*d^2 + 15*a^5*c^2*d^3)*x^3 - 8*(7*a^3*b^2*c^5 - 22*a^4*b*c^4*d - 93*a^5*c^3*d^2)*x^2 + 48*(a^4*b*c^5 + 2
1*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^3*x^5), 1/3840*(15*(7*b^5*c^5 - 25*a*b^4*c^4*d + 30*a^2*b^
3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(-a*c)*x^5*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*(384*a^5*c^5 - (10
5*a*b^4*c^5 - 340*a^2*b^3*c^4*d + 346*a^3*b^2*c^3*d^2 - 60*a^4*b*c^2*d^3 + 45*a^5*c*d^4)*x^4 + 2*(35*a^2*b^3*c
^5 - 111*a^3*b^2*c^4*d + 109*a^4*b*c^3*d^2 + 15*a^5*c^2*d^3)*x^3 - 8*(7*a^3*b^2*c^5 - 22*a^4*b*c^4*d - 93*a^5*
c^3*d^2)*x^2 + 48*(a^4*b*c^5 + 21*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^3*x^5)]

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giac [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((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^6,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.02, size = 967, normalized size = 3.42 \[ -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (45 a^{5} d^{5} x^{5} \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 )-75 a^{4} b c \,d^{4} x^{5} \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 )-150 a^{3} b^{2} c^{2} d^{3} x^{5} \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 )+450 a^{2} b^{3} c^{3} d^{2} x^{5} \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 )-375 a \,b^{4} c^{4} d \,x^{5} \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 )+105 b^{5} c^{5} x^{5} \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 )-90 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4} x^{4}+120 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3} x^{4}-692 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}+680 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d \,x^{4}-210 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4} x^{4}+60 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c \,d^{3} x^{3}+436 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{2} d^{2} x^{3}-444 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{3} d \,x^{3}+140 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{4} x^{3}+1488 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{2} d^{2} x^{2}+352 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{3} d \,x^{2}-112 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{4} x^{2}+2016 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{3} d x +96 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{4} x +768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^6,x)

[Out]

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^2*(45*a^5*d^5*x^5*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)-75*a^4*b*c*d^4*x^5*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)-150*a^3*b^2*c^2*d^3*x^5*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)+450*a^2*b^3
*c^3*d^2*x^5*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)-375*a*b^4*c^4*d*x^5*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)+105*b^5*c^5*x^5*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)-90*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*d^4*x^4+120*(a
*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c*d^3*x^4-692*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*
b^2*c^2*d^2*x^4+680*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^3*d*x^4-210*(a*c)^(1/2)*(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)*b^4*c^4*x^4+60*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c*d^3*x^3+436*(a*c)^(1/2)*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c^2*d^2*x^3-444*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^2*c^3*d*x
^3+140*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^4*x^3+1488*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1
/2)*a^4*c^2*d^2*x^2+352*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c^3*d*x^2-112*(a*c)^(1/2)*(b*d*x^2+a
*d*x+b*c*x+a*c)^(1/2)*a^2*b^2*c^4*x^2+2016*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c^3*d*x+96*(a*c)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c^4*x+768*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(a*c)^(1/2)*a^4*c^4)/(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)/x^5/(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((d*x+c)^(5/2)*(b*x+a)^(1/2)/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 details)Is a*d-b*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}}{x^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(1/2)*(c + d*x)^(5/2))/x^6,x)

[Out]

int(((a + b*x)^(1/2)*(c + d*x)^(5/2))/x^6, 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((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**6,x)

[Out]

Timed out

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