3.7.89 \(\int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx\) [689]

Optimal. Leaf size=317 \[ -\frac {d \left (15 b^3 c^3-839 a b^2 c^2 d+1785 a^2 b c d^2-945 a^3 d^3\right ) \sqrt {a+b x}}{192 a c^5 \sqrt {c+d x}}-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}+\frac {5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{11/2}} \]

[Out]

5/64*(-a*d+b*c)^2*(-63*a^2*d^2+14*a*b*c*d+b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(3/2
)/c^(11/2)-1/4*a*(b*x+a)^(3/2)/c/x^4/(d*x+c)^(1/2)-1/192*d*(-945*a^3*d^3+1785*a^2*b*c*d^2-839*a*b^2*c^2*d+15*b
^3*c^3)*(b*x+a)^(1/2)/a/c^5/(d*x+c)^(1/2)-1/24*a*(-9*a*d+11*b*c)*(b*x+a)^(1/2)/c^2/x^3/(d*x+c)^(1/2)-1/96*(-63
*a*d+59*b*c)*(-a*d+b*c)*(b*x+a)^(1/2)/c^3/x^2/(d*x+c)^(1/2)-1/192*(-a*d+b*c)*(315*a^2*d^2-322*a*b*c*d+15*b^2*c
^2)*(b*x+a)^(1/2)/a/c^4/x/(d*x+c)^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {100, 154, 156, 157, 12, 95, 214} \begin {gather*} -\frac {\sqrt {a+b x} \left (315 a^2 d^2-322 a b c d+15 b^2 c^2\right ) (b c-a d)}{192 a c^4 x \sqrt {c+d x}}+\frac {5 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{11/2}}-\frac {d \sqrt {a+b x} \left (-945 a^3 d^3+1785 a^2 b c d^2-839 a b^2 c^2 d+15 b^3 c^3\right )}{192 a c^5 \sqrt {c+d x}}-\frac {\sqrt {a+b x} (59 b c-63 a d) (b c-a d)}{96 c^3 x^2 \sqrt {c+d x}}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{24 c^2 x^3 \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/192*(d*(15*b^3*c^3 - 839*a*b^2*c^2*d + 1785*a^2*b*c*d^2 - 945*a^3*d^3)*Sqrt[a + b*x])/(a*c^5*Sqrt[c + d*x])
- (a*(11*b*c - 9*a*d)*Sqrt[a + b*x])/(24*c^2*x^3*Sqrt[c + d*x]) - ((59*b*c - 63*a*d)*(b*c - a*d)*Sqrt[a + b*x
])/(96*c^3*x^2*Sqrt[c + d*x]) - ((b*c - a*d)*(15*b^2*c^2 - 322*a*b*c*d + 315*a^2*d^2)*Sqrt[a + b*x])/(192*a*c^
4*x*Sqrt[c + d*x]) - (a*(a + b*x)^(3/2))/(4*c*x^4*Sqrt[c + d*x]) + (5*(b*c - a*d)^2*(b^2*c^2 + 14*a*b*c*d - 63
*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(11/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

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)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx &=-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}-\frac {\int \frac {\sqrt {a+b x} \left (-\frac {1}{2} a (11 b c-9 a d)-b (4 b c-3 a d) x\right )}{x^4 (c+d x)^{3/2}} \, dx}{4 c}\\ &=-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}-\frac {\int \frac {-\frac {1}{4} a (59 b c-63 a d) (b c-a d)-\frac {3}{2} b (8 b c-9 a d) (b c-a d) x}{x^3 \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{12 c^2}\\ &=-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}+\frac {\int \frac {\frac {1}{8} a (b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right )-\frac {1}{2} a b d (59 b c-63 a d) (b c-a d) x}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{24 a c^3}\\ &=-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}-\frac {\int \frac {\frac {15}{16} a (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )+\frac {1}{8} a b d (b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{24 a^2 c^4}\\ &=-\frac {d \left (15 b^3 c^3-839 a b^2 c^2 d+1785 a^2 b c d^2-945 a^3 d^3\right ) \sqrt {a+b x}}{192 a c^5 \sqrt {c+d x}}-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}+\frac {\int -\frac {15 a (b c-a d)^3 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{12 a^2 c^5 (b c-a d)}\\ &=-\frac {d \left (15 b^3 c^3-839 a b^2 c^2 d+1785 a^2 b c d^2-945 a^3 d^3\right ) \sqrt {a+b x}}{192 a c^5 \sqrt {c+d x}}-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}-\frac {\left (5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a c^5}\\ &=-\frac {d \left (15 b^3 c^3-839 a b^2 c^2 d+1785 a^2 b c d^2-945 a^3 d^3\right ) \sqrt {a+b x}}{192 a c^5 \sqrt {c+d x}}-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}-\frac {\left (5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )\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 c^5}\\ &=-\frac {d \left (15 b^3 c^3-839 a b^2 c^2 d+1785 a^2 b c d^2-945 a^3 d^3\right ) \sqrt {a+b x}}{192 a c^5 \sqrt {c+d x}}-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}+\frac {5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.64, size = 238, normalized size = 0.75 \begin {gather*} \frac {\sqrt {a+b x} \left (-15 b^3 c^3 x^3 (c+d x)+a b^2 c^2 x^2 \left (-118 c^2+337 c d x+839 d^2 x^2\right )-a^2 b c x \left (136 c^3-244 c^2 d x+637 c d^2 x^2+1785 d^3 x^3\right )+a^3 \left (-48 c^4+72 c^3 d x-126 c^2 d^2 x^2+315 c d^3 x^3+945 d^4 x^4\right )\right )}{192 a c^5 x^4 \sqrt {c+d x}}+\frac {5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*(-15*b^3*c^3*x^3*(c + d*x) + a*b^2*c^2*x^2*(-118*c^2 + 337*c*d*x + 839*d^2*x^2) - a^2*b*c*x*(13
6*c^3 - 244*c^2*d*x + 637*c*d^2*x^2 + 1785*d^3*x^3) + a^3*(-48*c^4 + 72*c^3*d*x - 126*c^2*d^2*x^2 + 315*c*d^3*
x^3 + 945*d^4*x^4)))/(192*a*c^5*x^4*Sqrt[c + d*x]) + (5*(b*c - a*d)^2*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*ArcT
anh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(11/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(981\) vs. \(2(273)=546\).
time = 0.07, size = 982, normalized size = 3.10

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \left (945 \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^{5} x^{5}-2100 \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^{4} x^{5}+1350 \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^{3} x^{5}-180 \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^{2} x^{5}-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 ) b^{4} c^{4} d \,x^{5}+945 \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} c \,d^{4} x^{4}-2100 \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^{2} d^{3} x^{4}+1350 \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^{3} d^{2} x^{4}-180 \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^{4} d \,x^{4}-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 ) b^{4} c^{5} x^{4}-1890 a^{3} d^{4} x^{4} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+3570 a^{2} b c \,d^{3} x^{4} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-1678 a \,b^{2} c^{2} d^{2} x^{4} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+30 b^{3} c^{3} d \,x^{4} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-630 a^{3} c \,d^{3} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+1274 a^{2} b \,c^{2} d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-674 a \,b^{2} c^{3} d \,x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+30 b^{3} c^{4} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+252 a^{3} c^{2} d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-488 a^{2} b \,c^{3} d \,x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+236 a \,b^{2} c^{4} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-144 a^{3} c^{3} d x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+272 a^{2} b \,c^{4} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+96 a^{3} c^{4} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{384 a \,c^{5} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{4} \sqrt {a c}\, \sqrt {d x +c}}\)	\(982\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(b*x+a)^(1/2)*(945*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^5*x^5-2100*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^4*x^5+1350*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^3*x^5-180*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^2*x^5-15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*b^4*c^4*d*x
^5+945*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*c*d^4*x^4-2100*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^2*d^3*x^4+1350*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^3*d^2*x^4-180*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^4*d*x^4-15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*b^4*c^5*x^4-1890*a^3*d^4*x^4
*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+3570*a^2*b*c*d^3*x^4*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-1678*a*b^2*c^2*d
^2*x^4*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+30*b^3*c^3*d*x^4*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-630*a^3*c*d^3*
x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+1274*a^2*b*c^2*d^2*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-674*a*b^2*c
^3*d*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+30*b^3*c^4*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+252*a^3*c^2*d^
2*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-488*a^2*b*c^3*d*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+236*a*b^2*c^
4*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-144*a^3*c^3*d*x*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+272*a^2*b*c^4*x*
(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+96*a^3*c^4*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2))/a/c^5/((d*x+c)*(b*x+a))^(1
/2)/x^4/(a*c)^(1/2)/(d*x+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)^(5/2)/x^5/(d*x+c)^(3/2),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 detail

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Fricas [A]
time = 6.69, size = 828, normalized size = 2.61 \begin {gather*} \left [-\frac {15 \, {\left ({\left (b^{4} c^{4} d + 12 \, a b^{3} c^{3} d^{2} - 90 \, a^{2} b^{2} c^{2} d^{3} + 140 \, a^{3} b c d^{4} - 63 \, a^{4} d^{5}\right )} x^{5} + {\left (b^{4} c^{5} + 12 \, a b^{3} c^{4} d - 90 \, a^{2} b^{2} c^{3} d^{2} + 140 \, a^{3} b c^{2} d^{3} - 63 \, a^{4} c d^{4}\right )} x^{4}\right )} \sqrt {a c} \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^{5} + {\left (15 \, a b^{3} c^{4} d - 839 \, a^{2} b^{2} c^{3} d^{2} + 1785 \, a^{3} b c^{2} d^{3} - 945 \, a^{4} c d^{4}\right )} x^{4} + {\left (15 \, a b^{3} c^{5} - 337 \, a^{2} b^{2} c^{4} d + 637 \, a^{3} b c^{3} d^{2} - 315 \, a^{4} c^{2} d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{5} - 122 \, a^{3} b c^{4} d + 63 \, a^{4} c^{3} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{5} - 9 \, a^{4} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (a^{2} c^{6} d x^{5} + a^{2} c^{7} x^{4}\right )}}, -\frac {15 \, {\left ({\left (b^{4} c^{4} d + 12 \, a b^{3} c^{3} d^{2} - 90 \, a^{2} b^{2} c^{2} d^{3} + 140 \, a^{3} b c d^{4} - 63 \, a^{4} d^{5}\right )} x^{5} + {\left (b^{4} c^{5} + 12 \, a b^{3} c^{4} d - 90 \, a^{2} b^{2} c^{3} d^{2} + 140 \, a^{3} b c^{2} d^{3} - 63 \, a^{4} c d^{4}\right )} x^{4}\right )} \sqrt {-a c} \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^{5} + {\left (15 \, a b^{3} c^{4} d - 839 \, a^{2} b^{2} c^{3} d^{2} + 1785 \, a^{3} b c^{2} d^{3} - 945 \, a^{4} c d^{4}\right )} x^{4} + {\left (15 \, a b^{3} c^{5} - 337 \, a^{2} b^{2} c^{4} d + 637 \, a^{3} b c^{3} d^{2} - 315 \, a^{4} c^{2} d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{5} - 122 \, a^{3} b c^{4} d + 63 \, a^{4} c^{3} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{5} - 9 \, a^{4} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (a^{2} c^{6} d x^{5} + a^{2} c^{7} x^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(15*((b^4*c^4*d + 12*a*b^3*c^3*d^2 - 90*a^2*b^2*c^2*d^3 + 140*a^3*b*c*d^4 - 63*a^4*d^5)*x^5 + (b^4*c^5
+ 12*a*b^3*c^4*d - 90*a^2*b^2*c^3*d^2 + 140*a^3*b*c^2*d^3 - 63*a^4*c*d^4)*x^4)*sqrt(a*c)*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^5 + (15*a*b^3*c^4*d - 839*a^2*b^2*c^3*d^2 + 1785*a^3*b*c^2*d^3 - 945*a^4*c*
d^4)*x^4 + (15*a*b^3*c^5 - 337*a^2*b^2*c^4*d + 637*a^3*b*c^3*d^2 - 315*a^4*c^2*d^3)*x^3 + 2*(59*a^2*b^2*c^5 -
122*a^3*b*c^4*d + 63*a^4*c^3*d^2)*x^2 + 8*(17*a^3*b*c^5 - 9*a^4*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^
6*d*x^5 + a^2*c^7*x^4), -1/384*(15*((b^4*c^4*d + 12*a*b^3*c^3*d^2 - 90*a^2*b^2*c^2*d^3 + 140*a^3*b*c*d^4 - 63*
a^4*d^5)*x^5 + (b^4*c^5 + 12*a*b^3*c^4*d - 90*a^2*b^2*c^3*d^2 + 140*a^3*b*c^2*d^3 - 63*a^4*c*d^4)*x^4)*sqrt(-a
*c)*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^5 + (15*a*b^3*c^4*d - 839*a^2*b^2*c^3*d^2 + 1785*a^3*b*c^2*d^3 - 945*a^4*c*d^4)
*x^4 + (15*a*b^3*c^5 - 337*a^2*b^2*c^4*d + 637*a^3*b*c^3*d^2 - 315*a^4*c^2*d^3)*x^3 + 2*(59*a^2*b^2*c^5 - 122*
a^3*b*c^4*d + 63*a^4*c^3*d^2)*x^2 + 8*(17*a^3*b*c^5 - 9*a^4*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^6*d*
x^5 + a^2*c^7*x^4)]

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Sympy [F(-1)] Timed out
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((b*x+a)**(5/2)/x**5/(d*x+c)**(3/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3762 vs. \(2 (273) = 546\).
time = 25.96, size = 3762, normalized size = 11.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*sqrt(b*x + a)/(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*c^5*abs(b)) +
5/64*(sqrt(b*d)*b^6*c^4 + 12*sqrt(b*d)*a*b^5*c^3*d - 90*sqrt(b*d)*a^2*b^4*c^2*d^2 + 140*sqrt(b*d)*a^3*b^3*c*d
^3 - 63*sqrt(b*d)*a^4*b^2*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^5*abs(b)) - 1/96*(15*sqrt(b*d)*b^20*c^11 - 575*sqrt
(b*d)*a*b^19*c^10*d + 5077*sqrt(b*d)*a^2*b^18*c^9*d^2 - 22277*sqrt(b*d)*a^3*b^17*c^8*d^3 + 59494*sqrt(b*d)*a^4
*b^16*c^7*d^4 - 105350*sqrt(b*d)*a^5*b^15*c^6*d^5 + 128506*sqrt(b*d)*a^6*b^14*c^5*d^6 - 109082*sqrt(b*d)*a^7*b
^13*c^4*d^7 + 63547*sqrt(b*d)*a^8*b^12*c^3*d^8 - 24299*sqrt(b*d)*a^9*b^11*c^2*d^9 + 5505*sqrt(b*d)*a^10*b^10*c
*d^10 - 561*sqrt(b*d)*a^11*b^9*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^18*c^10 + 3946*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^17*c^
9*d - 26165*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^16*c^8*d^2 + 773
04*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^15*c^7*d^3 - 118178*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^14*c^6*d^4 + 80188*sqrt(b*d)*(sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^13*c^5*d^5 + 22494*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^12*c^4*d^6 - 87560*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^11*c^3*d^7 + 70411*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^10*c^2*d^8 - 26262*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^9*c*d^9 + 3927*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^8*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
^16*c^9 - 11309*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^15*c^8*d + 532
92*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^14*c^7*d^2 - 97092*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^13*c^6*d^3 + 73162*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^12*c^5*d^4 - 21174*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^5*b^11*c^4*d^5 + 36540*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^10*c^3*d^6 - 71636*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^9*c^2*d^7 + 49683*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^8*c*d^8 - 11781*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^7*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^14
*c^8 + 17480*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^13*c^7*d - 53740*
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^12*c^6*d^2 + 51320*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^11*c^5*d^3 - 12238*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^10*c^4*d^4 - 6664*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^9*c^3*d^5 + 31892*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6*b^8*c^2*d^6 - 47160*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^7*c*d^7 + 19635*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^6*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^12*c^7 - 15625*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^11*c^6*d + 26
765*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^10*c^5*d^2 - 7185*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^9*c^4*d^3 - 1489*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^8*c^3*d^4 - 5683*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^7*c^2*d^5 + 24375*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^6*b^6*c*d^6 - 19635*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^5*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^10*c^6 + 7970*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^9*
c^5*d - 4917*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^8*c^4*d^2 - 40
68*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^7*c^3*d^3 - 3829*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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