3.8.100 \(\int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx\) [800]

Optimal. Leaf size=220 \[ \frac {5 (7 b c-3 a d) (b c-a d) \sqrt {c+d x}}{4 a^4 \sqrt {a+b x}}+\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}-\frac {5 \sqrt {c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2}} \]

[Out]

5/12*(-3*a*d+7*b*c)*(-a*d+b*c)*(d*x+c)^(3/2)/a^3/c/(b*x+a)^(3/2)+1/4*(-3*a*d+7*b*c)*(d*x+c)^(5/2)/a^2/c/x/(b*x
+a)^(3/2)-1/2*(d*x+c)^(7/2)/a/c/x^2/(b*x+a)^(3/2)-5/4*(-3*a*d+7*b*c)*(-a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/
a^(1/2)/(d*x+c)^(1/2))*c^(1/2)/a^(9/2)+5/4*(-3*a*d+7*b*c)*(-a*d+b*c)*(d*x+c)^(1/2)/a^4/(b*x+a)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 220, 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 = {98, 96, 95, 214} \begin {gather*} -\frac {5 \sqrt {c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2}}+\frac {5 \sqrt {c+d x} (7 b c-3 a d) (b c-a d)}{4 a^4 \sqrt {a+b x}}+\frac {5 (c+d x)^{3/2} (7 b c-3 a d) (b c-a d)}{12 a^3 c (a+b x)^{3/2}}+\frac {(c+d x)^{5/2} (7 b c-3 a d)}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(7*b*c - 3*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(4*a^4*Sqrt[a + b*x]) + (5*(7*b*c - 3*a*d)*(b*c - a*d)*(c + d*x)
^(3/2))/(12*a^3*c*(a + b*x)^(3/2)) + ((7*b*c - 3*a*d)*(c + d*x)^(5/2))/(4*a^2*c*x*(a + b*x)^(3/2)) - (c + d*x)
^(7/2)/(2*a*c*x^2*(a + b*x)^(3/2)) - (5*Sqrt[c]*(7*b*c - 3*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S
qrt[a]*Sqrt[c + d*x])])/(4*a^(9/2))

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

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 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^3 (a+b x)^{5/2}} \, dx &=-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}-\frac {\left (\frac {7 b c}{2}-\frac {3 a d}{2}\right ) \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx}{2 a c}\\ &=\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 (7 b c-3 a d) (b c-a d)) \int \frac {(c+d x)^{3/2}}{x (a+b x)^{5/2}} \, dx}{8 a^2 c}\\ &=\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 (7 b c-3 a d) (b c-a d)) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}} \, dx}{8 a^3}\\ &=\frac {5 (7 b c-3 a d) (b c-a d) \sqrt {c+d x}}{4 a^4 \sqrt {a+b x}}+\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 c (7 b c-3 a d) (b c-a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^4}\\ &=\frac {5 (7 b c-3 a d) (b c-a d) \sqrt {c+d x}}{4 a^4 \sqrt {a+b x}}+\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 c (7 b c-3 a d) (b c-a d)) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^4}\\ &=\frac {5 (7 b c-3 a d) (b c-a d) \sqrt {c+d x}}{4 a^4 \sqrt {a+b x}}+\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}-\frac {5 \sqrt {c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 10.23, size = 159, normalized size = 0.72 \begin {gather*} \frac {-3 a^{7/2} (c+d x)^{7/2}+\frac {1}{2} (7 b c-3 a d) x \left (3 a^{5/2} (c+d x)^{5/2}+5 (b c-a d) x \left (\sqrt {a} \sqrt {c+d x} (4 a c+3 b c x+a d x)-3 c^{3/2} (a+b x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )}{6 a^{9/2} c x^2 (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a^(7/2)*(c + d*x)^(7/2) + ((7*b*c - 3*a*d)*x*(3*a^(5/2)*(c + d*x)^(5/2) + 5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c
+ d*x]*(4*a*c + 3*b*c*x + a*d*x) - 3*c^(3/2)*(a + b*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c +
d*x])])))/2)/(6*a^(9/2)*c*x^2*(a + b*x)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(757\) vs. \(2(182)=364\).
time = 0.08, size = 758, normalized size = 3.45

method	result	size
default	\(-\frac {\sqrt {d x +c}\, \left (45 \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 \,d^{2} x^{4}-150 \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^{2} d \,x^{4}+105 \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^{3} x^{4}+90 \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^{2} x^{3}-300 \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 \,x^{3}+210 \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} x^{3}+45 \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^{2} x^{2}-150 \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 \,x^{2}+105 \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} x^{2}-32 a^{2} b \,d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+230 a \,b^{2} c d \,x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-210 b^{3} c^{2} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-48 a^{3} d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+316 a^{2} b c d \,x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-280 a \,b^{2} c^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+54 a^{3} c d x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-42 a^{2} b \,c^{2} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+12 a^{3} c^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{24 a^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{2} \sqrt {a c}\, \left (b x +a \right )^{\frac {3}{2}}}\)	\(758\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(d*x+c)^(1/2)*(45*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*d^2*x^4-150*
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^2*d*x^4+105*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^3*x^4+90*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^2*x^3-300*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*x^3+2
10*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*x^3+45*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^2*x^2-150*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*x^2+105*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*x^2-
32*a^2*b*d^2*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+230*a*b^2*c*d*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-210
*b^3*c^2*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-48*a^3*d^2*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+316*a^2*b*
c*d*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-280*a*b^2*c^2*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+54*a^3*c*d*x
*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-42*a^2*b*c^2*x*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+12*a^3*c^2*(a*c)^(1/2)
*((d*x+c)*(b*x+a))^(1/2))/a^4/((d*x+c)*(b*x+a))^(1/2)/x^2/(a*c)^(1/2)/(b*x+a)^(3/2)

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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^3/(b*x+a)^(5/2),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 = 2.23, size = 659, normalized size = 3.00 \begin {gather*} \left [\frac {15 \, {\left ({\left (7 \, b^{4} c^{2} - 10 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (7 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + {\left (7 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} x^{2}\right )} \sqrt {\frac {c}{a}} \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^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (6 \, a^{3} c^{2} - {\left (105 \, b^{3} c^{2} - 115 \, a b^{2} c d + 16 \, a^{2} b d^{2}\right )} x^{3} - 2 \, {\left (70 \, a b^{2} c^{2} - 79 \, a^{2} b c d + 12 \, a^{3} d^{2}\right )} x^{2} - 3 \, {\left (7 \, a^{2} b c^{2} - 9 \, a^{3} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac {15 \, {\left ({\left (7 \, b^{4} c^{2} - 10 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (7 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + {\left (7 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} x^{2}\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) - 2 \, {\left (6 \, a^{3} c^{2} - {\left (105 \, b^{3} c^{2} - 115 \, a b^{2} c d + 16 \, a^{2} b d^{2}\right )} x^{3} - 2 \, {\left (70 \, a b^{2} c^{2} - 79 \, a^{2} b c d + 12 \, a^{3} d^{2}\right )} x^{2} - 3 \, {\left (7 \, a^{2} b c^{2} - 9 \, a^{3} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(15*((7*b^4*c^2 - 10*a*b^3*c*d + 3*a^2*b^2*d^2)*x^4 + 2*(7*a*b^3*c^2 - 10*a^2*b^2*c*d + 3*a^3*b*d^2)*x^3
+ (7*a^2*b^2*c^2 - 10*a^3*b*c*d + 3*a^4*d^2)*x^2)*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*
x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) -
4*(6*a^3*c^2 - (105*b^3*c^2 - 115*a*b^2*c*d + 16*a^2*b*d^2)*x^3 - 2*(70*a*b^2*c^2 - 79*a^2*b*c*d + 12*a^3*d^2)
*x^2 - 3*(7*a^2*b*c^2 - 9*a^3*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2), 1/24
*(15*((7*b^4*c^2 - 10*a*b^3*c*d + 3*a^2*b^2*d^2)*x^4 + 2*(7*a*b^3*c^2 - 10*a^2*b^2*c*d + 3*a^3*b*d^2)*x^3 + (7
*a^2*b^2*c^2 - 10*a^3*b*c*d + 3*a^4*d^2)*x^2)*sqrt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt
(d*x + c)*sqrt(-c/a)/(b*c*d*x^2 + a*c^2 + (b*c^2 + a*c*d)*x)) - 2*(6*a^3*c^2 - (105*b^3*c^2 - 115*a*b^2*c*d +
16*a^2*b*d^2)*x^3 - 2*(70*a*b^2*c^2 - 79*a^2*b*c*d + 12*a^3*d^2)*x^2 - 3*(7*a^2*b*c^2 - 9*a^3*c*d)*x)*sqrt(b*x
+ a)*sqrt(d*x + c))/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2)]

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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**3/(b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1694 vs. \(2 (182) = 364\).
time = 7.07, size = 1694, normalized size = 7.70 \begin {gather*} -\frac {5 \, {\left (7 \, \sqrt {b d} b^{2} c^{3} {\left | b \right |} - 10 \, \sqrt {b d} a b c^{2} d {\left | b \right |} + 3 \, \sqrt {b d} a^{2} c d^{2} {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{4 \, \sqrt {-a b c d} a^{4} b} + \frac {11 \, \sqrt {b d} b^{8} c^{6} {\left | b \right |} - 53 \, \sqrt {b d} a b^{7} c^{5} d {\left | b \right |} + 102 \, \sqrt {b d} a^{2} b^{6} c^{4} d^{2} {\left | b \right |} - 98 \, \sqrt {b d} a^{3} b^{5} c^{3} d^{3} {\left | b \right |} + 47 \, \sqrt {b d} a^{4} b^{4} c^{2} d^{4} {\left | b \right |} - 9 \, \sqrt {b d} a^{5} b^{3} c d^{5} {\left | b \right |} - 33 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{6} c^{5} {\left | b \right |} + 56 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{4} d {\left | b \right |} + 14 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c^{3} d^{2} {\left | b \right |} - 64 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{3} c^{2} d^{3} {\left | b \right |} + 27 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{2} c d^{4} {\left | b \right |} + 33 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{4} c^{4} {\left | b \right |} - 5 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} c^{3} d {\left | b \right |} + 15 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - 27 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b c d^{3} {\left | b \right |} - 11 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{2} c^{3} {\left | b \right |} + 2 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b c^{2} d {\left | b \right |} + 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} c d^{2} {\left | b \right |}}{2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )}^{2} a^{4}} + \frac {4 \, {\left (9 \, \sqrt {b d} b^{6} c^{5} {\left | b \right |} - 38 \, \sqrt {b d} a b^{5} c^{4} d {\left | b \right |} + 62 \, \sqrt {b d} a^{2} b^{4} c^{3} d^{2} {\left | b \right |} - 48 \, \sqrt {b d} a^{3} b^{3} c^{2} d^{3} {\left | b \right |} + 17 \, \sqrt {b d} a^{4} b^{2} c d^{4} {\left | b \right |} - 2 \, \sqrt {b d} a^{5} b d^{5} {\left | b \right |} - 18 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{4} c^{4} {\left | b \right |} + 60 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{3} c^{3} d {\left | b \right |} - 72 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{2} c^{2} d^{2} {\left | b \right |} + 36 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b c d^{3} {\left | b \right |} - 6 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} d^{4} {\left | b \right |} + 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{2} c^{3} {\left | b \right |} - 18 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b c^{2} d {\left | b \right |} + 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} c d^{2} {\left | b \right |}\right )}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/4*(7*sqrt(b*d)*b^2*c^3*abs(b) - 10*sqrt(b*d)*a*b*c^2*d*abs(b) + 3*sqrt(b*d)*a^2*c*d^2*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) + 1/2*(11*sqrt(b*d)*b^8*c^6*abs(b) - 53*sqrt(b*d)*a*b^7*c^5*d*abs(b) + 102*sqrt(b*d)*a^2*b^6*c
^4*d^2*abs(b) - 98*sqrt(b*d)*a^3*b^5*c^3*d^3*abs(b) + 47*sqrt(b*d)*a^4*b^4*c^2*d^4*abs(b) - 9*sqrt(b*d)*a^5*b^
3*c*d^5*abs(b) - 33*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^6*c^5*abs(b)
+ 56*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^5*c^4*d*abs(b) + 14*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^4*c^3*d^2*abs(b) - 64*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^3*c^2*d^3*abs(b) + 27*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^2*c*d^4*abs(b) + 33*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^4*c^4*abs(b) - 5*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*b^3*c^3*d*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))^4*a^2*b^2*c^2*d^2*abs(b) - 27*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*c*d^3*abs(b) - 11*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^6*b^2*c^3*abs(b) + 2*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*
c^2*d*abs(b) + 9*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*c*d^2*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)^2*a^4) + 4/3*(9*sqrt(b*d)*b^6*c^5*abs(b) - 38*sqrt(b*d)*a*b^5*c^4*d*ab
s(b) + 62*sqrt(b*d)*a^2*b^4*c^3*d^2*abs(b) - 48*sqrt(b*d)*a^3*b^3*c^2*d^3*abs(b) + 17*sqrt(b*d)*a^4*b^2*c*d^4*
abs(b) - 2*sqrt(b*d)*a^5*b*d^5*abs(b) - 18*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^2*b^4*c^4*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))^2*a*b^3
*c^3*d*abs(b) - 72*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^2*c^2*d^2
*abs(b) + 36*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*c*d^3*abs(b) -
6*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*d^4*abs(b) + 9*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^2*c^3*abs(b) - 18*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*c^2*d*abs(b) + 9*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^2*c*d^2*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)^3*a^4)

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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^3\,{\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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