3.8.75 \(\int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx\) [775]

Optimal. Leaf size=318 \[ \frac {b \left (945 b^3 c^3-1785 a b^2 c^2 d+839 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c+d x}}{192 a^5 c \sqrt {a+b x}}+\frac {c (9 b c-11 a d) \sqrt {c+d x}}{24 a^2 x^3 \sqrt {a+b x}}-\frac {(63 b c-59 a d) (b c-a d) \sqrt {c+d x}}{96 a^3 x^2 \sqrt {a+b x}}+\frac {(b c-a d) \left (315 b^2 c^2-322 a b c d+15 a^2 d^2\right ) \sqrt {c+d x}}{192 a^4 c x \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}-\frac {5 (b c-a d)^2 \left (63 b^2 c^2-14 a b c d-a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{11/2} c^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 318, 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 {c+d x} (63 b c-59 a d) (b c-a d)}{96 a^3 x^2 \sqrt {a+b x}}+\frac {c \sqrt {c+d x} (9 b c-11 a d)}{24 a^2 x^3 \sqrt {a+b x}}-\frac {5 \left (-a^2 d^2-14 a b c d+63 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^{11/2} c^{3/2}}+\frac {\sqrt {c+d x} \left (15 a^2 d^2-322 a b c d+315 b^2 c^2\right ) (b c-a d)}{192 a^4 c x \sqrt {a+b x}}+\frac {b \sqrt {c+d x} \left (-15 a^3 d^3+839 a^2 b c d^2-1785 a b^2 c^2 d+945 b^3 c^3\right )}{192 a^5 c \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.70, size = 241, normalized size = 0.76 \begin {gather*} \frac {\sqrt {c+d x} \left (945 b^4 c^3 x^4+105 a b^3 c^2 x^3 (3 c-17 d x)+a^2 b^2 c x^2 \left (-126 c^2-637 c d x+839 d^2 x^2\right )+a^3 b x \left (72 c^3+244 c^2 d x+337 c d^2 x^2-15 d^3 x^3\right )-a^4 \left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )\right )}{192 a^5 c x^4 \sqrt {a+b x}}+\frac {5 (b c-a d)^2 \left (-63 b^2 c^2+14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{64 a^{11/2} c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x]*(945*b^4*c^3*x^4 + 105*a*b^3*c^2*x^3*(3*c - 17*d*x) + a^2*b^2*c*x^2*(-126*c^2 - 637*c*d*x + 839
*d^2*x^2) + a^3*b*x*(72*c^3 + 244*c^2*d*x + 337*c*d^2*x^2 - 15*d^3*x^3) - a^4*(48*c^3 + 136*c^2*d*x + 118*c*d^
2*x^2 + 15*d^3*x^3)))/(192*a^5*c*x^4*Sqrt[a + b*x]) + (5*(b*c - a*d)^2*(-63*b^2*c^2 + 14*a*b*c*d + a^2*d^2)*Ar
cTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(64*a^(11/2)*c^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(981\) vs. \(2(274)=548\).
time = 0.07, size = 982, normalized size = 3.09

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(d*x+c)^(1/2)*(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)*a^4*b*d^4*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^3*b^2*c*d^3*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^3*c^2*d^2*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*b^4*c^3*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)*b^5*c^4*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)*a^5*d^4*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^4*b*c*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^3*b^2*c^2*d^2*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^2*b^3
*c^3*d*x^4-945*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*b^4*c^4*x^4-30*a^3*b*d^3*x^4*
(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+1678*a^2*b^2*c*d^2*x^4*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-3570*a*b^3*c^2*
d*x^4*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+1890*b^4*c^3*x^4*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-30*a^4*d^3*x^3*
(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+674*a^3*b*c*d^2*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-1274*a^2*b^2*c^2*d
*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+630*a*b^3*c^3*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-236*a^4*c*d^2*x
^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+488*a^3*b*c^2*d*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-252*a^2*b^2*c^3
*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-272*a^4*c^2*d*x*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+144*a^3*b*c^3*x*(
a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-96*a^4*c^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2))/a^5/c/((d*x+c)*(b*x+a))^(1/
2)/x^4/(a*c)^(1/2)/(b*x+a)^(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)^(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 = 4.44, size = 836, normalized size = 2.63 \begin {gather*} \left [-\frac {15 \, {\left ({\left (63 \, b^{5} c^{4} - 140 \, a b^{4} c^{3} d + 90 \, a^{2} b^{3} c^{2} d^{2} - 12 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{5} + {\left (63 \, a b^{4} c^{4} - 140 \, a^{2} b^{3} c^{3} d + 90 \, a^{3} b^{2} c^{2} d^{2} - 12 \, a^{4} b c d^{3} - a^{5} 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^{5} c^{4} - {\left (945 \, a b^{4} c^{4} - 1785 \, a^{2} b^{3} c^{3} d + 839 \, a^{3} b^{2} c^{2} d^{2} - 15 \, a^{4} b c d^{3}\right )} x^{4} - {\left (315 \, a^{2} b^{3} c^{4} - 637 \, a^{3} b^{2} c^{3} d + 337 \, a^{4} b c^{2} d^{2} - 15 \, a^{5} c d^{3}\right )} x^{3} + 2 \, {\left (63 \, a^{3} b^{2} c^{4} - 122 \, a^{4} b c^{3} d + 59 \, a^{5} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (9 \, a^{4} b c^{4} - 17 \, a^{5} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (a^{6} b c^{2} x^{5} + a^{7} c^{2} x^{4}\right )}}, \frac {15 \, {\left ({\left (63 \, b^{5} c^{4} - 140 \, a b^{4} c^{3} d + 90 \, a^{2} b^{3} c^{2} d^{2} - 12 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{5} + {\left (63 \, a b^{4} c^{4} - 140 \, a^{2} b^{3} c^{3} d + 90 \, a^{3} b^{2} c^{2} d^{2} - 12 \, a^{4} b c d^{3} - a^{5} 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^{5} c^{4} - {\left (945 \, a b^{4} c^{4} - 1785 \, a^{2} b^{3} c^{3} d + 839 \, a^{3} b^{2} c^{2} d^{2} - 15 \, a^{4} b c d^{3}\right )} x^{4} - {\left (315 \, a^{2} b^{3} c^{4} - 637 \, a^{3} b^{2} c^{3} d + 337 \, a^{4} b c^{2} d^{2} - 15 \, a^{5} c d^{3}\right )} x^{3} + 2 \, {\left (63 \, a^{3} b^{2} c^{4} - 122 \, a^{4} b c^{3} d + 59 \, a^{5} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (9 \, a^{4} b c^{4} - 17 \, a^{5} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (a^{6} b c^{2} x^{5} + a^{7} c^{2} x^{4}\right )}}\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)^(3/2),x, algorithm="fricas")

[Out]

[-1/768*(15*((63*b^5*c^4 - 140*a*b^4*c^3*d + 90*a^2*b^3*c^2*d^2 - 12*a^3*b^2*c*d^3 - a^4*b*d^4)*x^5 + (63*a*b^
4*c^4 - 140*a^2*b^3*c^3*d + 90*a^3*b^2*c^2*d^2 - 12*a^4*b*c*d^3 - a^5*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^5*c^4 - (945*a*b^4*c^4 - 1785*a^2*b^3*c^3*d + 839*a^3*b^2*c^2*d^2 - 15*a^4*b*c*
d^3)*x^4 - (315*a^2*b^3*c^4 - 637*a^3*b^2*c^3*d + 337*a^4*b*c^2*d^2 - 15*a^5*c*d^3)*x^3 + 2*(63*a^3*b^2*c^4 -
122*a^4*b*c^3*d + 59*a^5*c^2*d^2)*x^2 - 8*(9*a^4*b*c^4 - 17*a^5*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*b*
c^2*x^5 + a^7*c^2*x^4), 1/384*(15*((63*b^5*c^4 - 140*a*b^4*c^3*d + 90*a^2*b^3*c^2*d^2 - 12*a^3*b^2*c*d^3 - a^4
*b*d^4)*x^5 + (63*a*b^4*c^4 - 140*a^2*b^3*c^3*d + 90*a^3*b^2*c^2*d^2 - 12*a^4*b*c*d^3 - a^5*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^5*c^4 - (945*a*b^4*c^4 - 1785*a^2*b^3*c^3*d + 839*a^3*b^2*c^2*d^2 - 15*a^4*b*c*d^3)*
x^4 - (315*a^2*b^3*c^4 - 637*a^3*b^2*c^3*d + 337*a^4*b*c^2*d^2 - 15*a^5*c*d^3)*x^3 + 2*(63*a^3*b^2*c^4 - 122*a
^4*b*c^3*d + 59*a^5*c^2*d^2)*x^2 - 8*(9*a^4*b*c^4 - 17*a^5*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*b*c^2*x
^5 + a^7*c^2*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((d*x+c)**(5/2)/x**5/(b*x+a)**(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. 3947 vs. \(2 (274) = 548\).
time = 33.43, size = 3947, normalized size = 12.41 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(sqrt(b*d)*b^4*c^3*abs(b) - 3*sqrt(b*d)*a*b^3*c^2*d*abs(b) + 3*sqrt(b*d)*a^2*b^2*c*d^2*abs(b) - sqrt(b*d)*a^
3*b*d^3*abs(b))/((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)*a^5) - 5/
64*(63*sqrt(b*d)*b^4*c^4*abs(b) - 140*sqrt(b*d)*a*b^3*c^3*d*abs(b) + 90*sqrt(b*d)*a^2*b^2*c^2*d^2*abs(b) - 12*
sqrt(b*d)*a^3*b*c*d^3*abs(b) - sqrt(b*d)*a^4*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^5*b*c) + 1/96*(561*sqrt(b*d)*
b^18*c^11*abs(b) - 5505*sqrt(b*d)*a*b^17*c^10*d*abs(b) + 24299*sqrt(b*d)*a^2*b^16*c^9*d^2*abs(b) - 63547*sqrt(
b*d)*a^3*b^15*c^8*d^3*abs(b) + 109082*sqrt(b*d)*a^4*b^14*c^7*d^4*abs(b) - 128506*sqrt(b*d)*a^5*b^13*c^6*d^5*ab
s(b) + 105350*sqrt(b*d)*a^6*b^12*c^5*d^6*abs(b) - 59494*sqrt(b*d)*a^7*b^11*c^4*d^7*abs(b) + 22277*sqrt(b*d)*a^
8*b^10*c^3*d^8*abs(b) - 5077*sqrt(b*d)*a^9*b^9*c^2*d^9*abs(b) + 575*sqrt(b*d)*a^10*b^8*c*d^10*abs(b) - 15*sqrt
(b*d)*a^11*b^7*d^11*abs(b) - 3927*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*
b^16*c^10*abs(b) + 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*b^15*c^
9*d*abs(b) - 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^2*b^14*c^8*d^
2*abs(b) + 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^3*b^13*c^7*d^3*
abs(b) - 22494*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^12*c^6*d^4*ab
s(b) - 80188*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^11*c^5*d^5*abs(
b) + 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^6*b^10*c^4*d^6*abs(b
) - 77304*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^9*c^3*d^7*abs(b) +
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^8*b^8*c^2*d^8*abs(b) - 39
46*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^7*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^6*d^10*abs(b) + 11781*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^14*c^9*abs(b) - 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*b^13*c^8*d*abs(b) + 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^2*b^12*c^7*d^2*abs(b) - 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^3*b^11*c^6*d^3*abs(b) + 21174*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^10*c^5*d^4*abs(b) - 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^5*b^9*c^4*d^5*abs(b) + 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^6*b^8*c^3*d^6*abs(b) - 53292*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^7*c^2*d^7*abs(b) + 11309*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^8*b^6*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^5*d^9*abs(b) - 19635*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^6*b^12*c^8*abs(b) + 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*b^11*c^7*d*abs(b) - 31892*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^10*c^6*d^2*abs(b) + 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^3*b^9*c^5*d^3*abs(b) + 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^8*c^4*d^4*abs(b) - 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^5
*b^7*c^3*d^5*abs(b) + 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^6*b^
6*c^2*d^6*abs(b) - 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^7*b^5*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^4*d^8*abs(
b) + 19635*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^10*c^7*abs(b) - 24375
*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^9*c^6*d*abs(b) + 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^2*b^8*c^5*d^2*abs(b) + 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^3*b^7*c^4*d^3*abs(b) + 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^4*b^6*c^3*d^4*abs(b) - 26765*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^5*c^2*d^5*abs(b) + 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^6*b^4*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...

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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\,{\left (a+b\,x\right )}^{3/2}} \,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)^(3/2)),x)

[Out]

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

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