3.7.96 \(\int \frac {(a+b x)^{5/2}}{x^3 (c+d x)^{5/2}} \, dx\) [696]

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

[Out]

5/12*(-7*a*d+3*b*c)*(-a*d+b*c)*(b*x+a)^(3/2)/a/c^3/(d*x+c)^(3/2)-1/4*(-7*a*d+3*b*c)*(b*x+a)^(5/2)/a/c^2/x/(d*x
+c)^(3/2)-1/2*(b*x+a)^(7/2)/a/c/x^2/(d*x+c)^(3/2)-5/4*(-7*a*d+3*b*c)*(-a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/
a^(1/2)/(d*x+c)^(1/2))*a^(1/2)/c^(9/2)+5/4*(-7*a*d+3*b*c)*(-a*d+b*c)*(b*x+a)^(1/2)/c^4/(d*x+c)^(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 {a} (3 b c-7 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 c^{9/2}}+\frac {5 \sqrt {a+b x} (3 b c-7 a d) (b c-a d)}{4 c^4 \sqrt {c+d x}}+\frac {5 (a+b x)^{3/2} (3 b c-7 a d) (b c-a d)}{12 a c^3 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2} (3 b c-7 a d)}{4 a c^2 x (c+d x)^{3/2}}-\frac {(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(3*b*c - 7*a*d)*(b*c - a*d)*(a + b*x)^(3/2))/(12*a*c^3*(c + d*x)^(3/2)) - ((3*b*c - 7*a*d)*(a + b*x)^(5/2))
/(4*a*c^2*x*(c + d*x)^(3/2)) - (a + b*x)^(7/2)/(2*a*c*x^2*(c + d*x)^(3/2)) + (5*(3*b*c - 7*a*d)*(b*c - a*d)*Sq
rt[a + b*x])/(4*c^4*Sqrt[c + d*x]) - (5*Sqrt[a]*(3*b*c - 7*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S
qrt[a]*Sqrt[c + d*x])])/(4*c^(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 {(a+b x)^{5/2}}{x^3 (c+d x)^{5/2}} \, dx &=-\frac {(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}-\frac {\left (-\frac {3 b c}{2}+\frac {7 a d}{2}\right ) \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx}{2 a c}\\ &=-\frac {(3 b c-7 a d) (a+b x)^{5/2}}{4 a c^2 x (c+d x)^{3/2}}-\frac {(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}+\frac {(5 (3 b c-7 a d) (b c-a d)) \int \frac {(a+b x)^{3/2}}{x (c+d x)^{5/2}} \, dx}{8 a c^2}\\ &=\frac {5 (3 b c-7 a d) (b c-a d) (a+b x)^{3/2}}{12 a c^3 (c+d x)^{3/2}}-\frac {(3 b c-7 a d) (a+b x)^{5/2}}{4 a c^2 x (c+d x)^{3/2}}-\frac {(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}+\frac {(5 (3 b c-7 a d) (b c-a d)) \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx}{8 c^3}\\ &=\frac {5 (3 b c-7 a d) (b c-a d) (a+b x)^{3/2}}{12 a c^3 (c+d x)^{3/2}}-\frac {(3 b c-7 a d) (a+b x)^{5/2}}{4 a c^2 x (c+d x)^{3/2}}-\frac {(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}+\frac {5 (3 b c-7 a d) (b c-a d) \sqrt {a+b x}}{4 c^4 \sqrt {c+d x}}+\frac {(5 a (3 b c-7 a d) (b c-a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 c^4}\\ &=\frac {5 (3 b c-7 a d) (b c-a d) (a+b x)^{3/2}}{12 a c^3 (c+d x)^{3/2}}-\frac {(3 b c-7 a d) (a+b x)^{5/2}}{4 a c^2 x (c+d x)^{3/2}}-\frac {(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}+\frac {5 (3 b c-7 a d) (b c-a d) \sqrt {a+b x}}{4 c^4 \sqrt {c+d x}}+\frac {(5 a (3 b c-7 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 c^4}\\ &=\frac {5 (3 b c-7 a d) (b c-a d) (a+b x)^{3/2}}{12 a c^3 (c+d x)^{3/2}}-\frac {(3 b c-7 a d) (a+b x)^{5/2}}{4 a c^2 x (c+d x)^{3/2}}-\frac {(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}+\frac {5 (3 b c-7 a d) (b c-a d) \sqrt {a+b x}}{4 c^4 \sqrt {c+d x}}-\frac {5 \sqrt {a} (3 b c-7 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 c^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 10.21, size = 159, normalized size = 0.72 \begin {gather*} \frac {-3 c^{7/2} (a+b x)^{7/2}-\frac {1}{2} (3 b c-7 a d) x \left (3 c^{5/2} (a+b x)^{5/2}-5 (b c-a d) x \left (\sqrt {c} \sqrt {a+b x} (4 a c+b c x+3 a d x)-3 a^{3/2} (c+d x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )}{6 a c^{9/2} x^2 (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*c^(7/2)*(a + b*x)^(7/2) - ((3*b*c - 7*a*d)*x*(3*c^(5/2)*(a + b*x)^(5/2) - 5*(b*c - a*d)*x*(Sqrt[c]*Sqrt[a
+ b*x]*(4*a*c + b*c*x + 3*a*d*x) - 3*a^(3/2)*(c + d*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c +
d*x])])))/2)/(6*a*c^(9/2)*x^2*(c + d*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 {b x +a}\, \left (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^{3} d^{4} 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^{2} b c \,d^{3} x^{4}+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 \,b^{2} c^{2} d^{2} x^{4}+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^{3} c \,d^{3} 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 \,c^{2} d^{2} x^{3}+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 \,b^{2} c^{3} d \,x^{3}+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^{3} c^{2} 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^{2} b \,c^{3} d \,x^{2}+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 \,b^{2} c^{4} x^{2}-210 a^{2} d^{3} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+230 a b c \,d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-32 b^{2} c^{2} d \,x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-280 a^{2} c \,d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+316 a b \,c^{2} d \,x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-48 b^{2} c^{3} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-42 a^{2} c^{2} d x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+54 a b \,c^{3} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+12 a^{2} c^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{24 c^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{2} \sqrt {a c}\, \left (d x +c \right )^{\frac {3}{2}}}\)	\(758\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(b*x+a)^(1/2)*(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^3*d^4*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^2*b*c*d^3*x^4+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*b^2*c^2*d^2*x^4+210*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*c*d^3*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*c^2*d^2*x^3+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*b^2*c^3*d*x^3+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^3*c^2*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^2*b*c^3*d*x^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*b^2*c^4*x^2-
210*a^2*d^3*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+230*a*b*c*d^2*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-32*b
^2*c^2*d*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-280*a^2*c*d^2*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+316*a*b
*c^2*d*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-48*b^2*c^3*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-42*a^2*c^2*d
*x*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+54*a*b*c^3*x*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+12*a^2*c^3*(a*c)^(1/2)
*((d*x+c)*(b*x+a))^(1/2))/c^4/((d*x+c)*(b*x+a))^(1/2)/x^2/(a*c)^(1/2)/(d*x+c)^(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((b*x+a)^(5/2)/x^3/(d*x+c)^(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 = 3.65, size = 659, normalized size = 3.00 \begin {gather*} \left [\frac {15 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} x^{3} + {\left (3 \, b^{2} c^{4} - 10 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {\frac {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^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (6 \, a^{2} c^{3} - {\left (16 \, b^{2} c^{2} d - 115 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} x^{3} - 2 \, {\left (12 \, b^{2} c^{3} - 79 \, a b c^{2} d + 70 \, a^{2} c d^{2}\right )} x^{2} + 3 \, {\left (9 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (c^{4} d^{2} x^{4} + 2 \, c^{5} d x^{3} + c^{6} x^{2}\right )}}, \frac {15 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} x^{3} + {\left (3 \, b^{2} c^{4} - 10 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{c}} \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 {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - 2 \, {\left (6 \, a^{2} c^{3} - {\left (16 \, b^{2} c^{2} d - 115 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} x^{3} - 2 \, {\left (12 \, b^{2} c^{3} - 79 \, a b c^{2} d + 70 \, a^{2} c d^{2}\right )} x^{2} + 3 \, {\left (9 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (c^{4} d^{2} x^{4} + 2 \, c^{5} d x^{3} + c^{6} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(15*((3*b^2*c^2*d^2 - 10*a*b*c*d^3 + 7*a^2*d^4)*x^4 + 2*(3*b^2*c^3*d - 10*a*b*c^2*d^2 + 7*a^2*c*d^3)*x^3
+ (3*b^2*c^4 - 10*a*b*c^3*d + 7*a^2*c^2*d^2)*x^2)*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^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) -
4*(6*a^2*c^3 - (16*b^2*c^2*d - 115*a*b*c*d^2 + 105*a^2*d^3)*x^3 - 2*(12*b^2*c^3 - 79*a*b*c^2*d + 70*a^2*c*d^2)
*x^2 + 3*(9*a*b*c^3 - 7*a^2*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^4*d^2*x^4 + 2*c^5*d*x^3 + c^6*x^2), 1/24
*(15*((3*b^2*c^2*d^2 - 10*a*b*c*d^3 + 7*a^2*d^4)*x^4 + 2*(3*b^2*c^3*d - 10*a*b*c^2*d^2 + 7*a^2*c*d^3)*x^3 + (3
*b^2*c^4 - 10*a*b*c^3*d + 7*a^2*c^2*d^2)*x^2)*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt
(d*x + c)*sqrt(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) - 2*(6*a^2*c^3 - (16*b^2*c^2*d - 115*a*b*c*d^2 +
105*a^2*d^3)*x^3 - 2*(12*b^2*c^3 - 79*a*b*c^2*d + 70*a^2*c*d^2)*x^2 + 3*(9*a*b*c^3 - 7*a^2*c^2*d)*x)*sqrt(b*x
+ a)*sqrt(d*x + c))/(c^4*d^2*x^4 + 2*c^5*d*x^3 + c^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((b*x+a)**(5/2)/x**3/(d*x+c)**(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. 1278 vs. \(2 (182) = 364\).
time = 3.35, size = 1278, normalized size = 5.81 \begin {gather*} \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (2 \, b^{6} c^{7} d^{2} {\left | b \right |} - 13 \, a b^{5} c^{6} d^{3} {\left | b \right |} + 20 \, a^{2} b^{4} c^{5} d^{4} {\left | b \right |} - 9 \, a^{3} b^{3} c^{4} d^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c^{9} d - a b^{2} c^{8} d^{2}} + \frac {3 \, {\left (b^{7} c^{8} d {\left | b \right |} - 6 \, a b^{6} c^{7} d^{2} {\left | b \right |} + 12 \, a^{2} b^{5} c^{6} d^{3} {\left | b \right |} - 10 \, a^{3} b^{4} c^{5} d^{4} {\left | b \right |} + 3 \, a^{4} b^{3} c^{4} d^{5} {\left | b \right |}\right )}}{b^{3} c^{9} d - a b^{2} c^{8} d^{2}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (3 \, \sqrt {b d} a b^{4} c^{2} - 10 \, \sqrt {b d} a^{2} b^{3} c d + 7 \, \sqrt {b d} a^{3} b^{2} d^{2}\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} b c^{4} {\left | b \right |}} - \frac {9 \, \sqrt {b d} a b^{10} c^{5} - 47 \, \sqrt {b d} a^{2} b^{9} c^{4} d + 98 \, \sqrt {b d} a^{3} b^{8} c^{3} d^{2} - 102 \, \sqrt {b d} a^{4} b^{7} c^{2} d^{3} + 53 \, \sqrt {b d} a^{5} b^{6} c d^{4} - 11 \, \sqrt {b d} a^{6} b^{5} d^{5} - 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 b^{8} c^{4} + 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^{2} b^{7} c^{3} d - 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^{3} b^{6} c^{2} d^{2} - 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^{4} b^{5} c d^{3} + 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} a^{5} b^{4} d^{4} + 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 b^{6} c^{3} - 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^{5} c^{2} d + 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^{3} b^{4} c d^{2} - 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} a^{4} b^{3} d^{3} - 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 b^{4} c^{2} - 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^{2} b^{3} c d + 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} a^{3} b^{2} d^{2}}{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} c^{4} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(b*x + a)*((2*b^6*c^7*d^2*abs(b) - 13*a*b^5*c^6*d^3*abs(b) + 20*a^2*b^4*c^5*d^4*abs(b) - 9*a^3*b^3*c^4
*d^5*abs(b))*(b*x + a)/(b^3*c^9*d - a*b^2*c^8*d^2) + 3*(b^7*c^8*d*abs(b) - 6*a*b^6*c^7*d^2*abs(b) + 12*a^2*b^5
*c^6*d^3*abs(b) - 10*a^3*b^4*c^5*d^4*abs(b) + 3*a^4*b^3*c^4*d^5*abs(b))/(b^3*c^9*d - a*b^2*c^8*d^2))/(b^2*c +
(b*x + a)*b*d - a*b*d)^(3/2) - 5/4*(3*sqrt(b*d)*a*b^4*c^2 - 10*sqrt(b*d)*a^2*b^3*c*d + 7*sqrt(b*d)*a^3*b^2*d^2
)*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)*b*c^4*abs(b)) - 1/2*(9*sqrt(b*d)*a*b^10*c^5 - 47*sqrt(b*d)*a^2*b^9*c^4*d + 98*sqrt(b*d
)*a^3*b^8*c^3*d^2 - 102*sqrt(b*d)*a^4*b^7*c^2*d^3 + 53*sqrt(b*d)*a^5*b^6*c*d^4 - 11*sqrt(b*d)*a^6*b^5*d^5 - 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*b^8*c^4 + 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^2*b^7*c^3*d - 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^3*b^6*c^2*d^2 - 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^4*b^5*c*d^3 + 33*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^4*d^4 + 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*
b^6*c^3 - 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^5*c^2*d + 5*sqr
t(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^4*c*d^2 - 33*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^3*d^3 - 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*b^4*c^2 - 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^2*b^3*c*d + 11*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^2*d^2)/((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)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)^2*c^4*abs(b))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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