\(\int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx\) [796]

Optimal result

Integrand size = 20, antiderivative size = 222 \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {5 d (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^4}+\frac {5 d (3 b c-7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{6 b^3 (b c-a d)}-\frac {2 (3 b c-7 a d) (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt {a+b x}}+\frac {2 a (c+d x)^{7/2}}{3 b (b c-a d) (a+b x)^{3/2}}+\frac {5 \sqrt {d} (3 b c-7 a d) (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{9/2}} \]

[Out]

2/3*a*(d*x+c)^(7/2)/b/(-a*d+b*c)/(b*x+a)^(3/2)+5/4*(-7*a*d+3*b*c)*(-a*d+b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(
1/2)/(d*x+c)^(1/2))*d^(1/2)/b^(9/2)-2/3*(-7*a*d+3*b*c)*(d*x+c)^(5/2)/b^2/(-a*d+b*c)/(b*x+a)^(1/2)+5/6*d*(-7*a*
d+3*b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/b^3/(-a*d+b*c)+5/4*d*(-7*a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^4

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {79, 49, 52, 65, 223, 212} \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {5 \sqrt {d} (3 b c-7 a d) (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{9/2}}+\frac {5 d \sqrt {a+b x} \sqrt {c+d x} (3 b c-7 a d)}{4 b^4}+\frac {5 d \sqrt {a+b x} (c+d x)^{3/2} (3 b c-7 a d)}{6 b^3 (b c-a d)}-\frac {2 (c+d x)^{5/2} (3 b c-7 a d)}{3 b^2 \sqrt {a+b x} (b c-a d)}+\frac {2 a (c+d x)^{7/2}}{3 b (a+b x)^{3/2} (b c-a d)} \]

[In]

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

[Out]

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

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 a (c+d x)^{7/2}}{3 b (b c-a d) (a+b x)^{3/2}}+\frac {(3 b c-7 a d) \int \frac {(c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx}{3 b (b c-a d)} \\ & = -\frac {2 (3 b c-7 a d) (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt {a+b x}}+\frac {2 a (c+d x)^{7/2}}{3 b (b c-a d) (a+b x)^{3/2}}+\frac {(5 d (3 b c-7 a d)) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{3 b^2 (b c-a d)} \\ & = \frac {5 d (3 b c-7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{6 b^3 (b c-a d)}-\frac {2 (3 b c-7 a d) (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt {a+b x}}+\frac {2 a (c+d x)^{7/2}}{3 b (b c-a d) (a+b x)^{3/2}}+\frac {(5 d (3 b c-7 a d)) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{4 b^3} \\ & = \frac {5 d (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^4}+\frac {5 d (3 b c-7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{6 b^3 (b c-a d)}-\frac {2 (3 b c-7 a d) (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt {a+b x}}+\frac {2 a (c+d x)^{7/2}}{3 b (b c-a d) (a+b x)^{3/2}}+\frac {(5 d (3 b c-7 a d) (b c-a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^4} \\ & = \frac {5 d (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^4}+\frac {5 d (3 b c-7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{6 b^3 (b c-a d)}-\frac {2 (3 b c-7 a d) (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt {a+b x}}+\frac {2 a (c+d x)^{7/2}}{3 b (b c-a d) (a+b x)^{3/2}}+\frac {(5 d (3 b c-7 a d) (b c-a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b^5} \\ & = \frac {5 d (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^4}+\frac {5 d (3 b c-7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{6 b^3 (b c-a d)}-\frac {2 (3 b c-7 a d) (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt {a+b x}}+\frac {2 a (c+d x)^{7/2}}{3 b (b c-a d) (a+b x)^{3/2}}+\frac {(5 d (3 b c-7 a d) (b c-a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 b^5} \\ & = \frac {5 d (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^4}+\frac {5 d (3 b c-7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{6 b^3 (b c-a d)}-\frac {2 (3 b c-7 a d) (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt {a+b x}}+\frac {2 a (c+d x)^{7/2}}{3 b (b c-a d) (a+b x)^{3/2}}+\frac {5 \sqrt {d} (3 b c-7 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{9/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.55 \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {2 \sqrt {c+d x} \left (a b^3 (c+d x)^3-\frac {(3 b c-7 a d) (b c-a d)^2 (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {1}{2},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{3 b^4 (b c-a d) (a+b x)^{3/2}} \]

[In]

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

[Out]

(2*Sqrt[c + d*x]*(a*b^3*(c + d*x)^3 - ((3*b*c - 7*a*d)*(b*c - a*d)^2*(a + b*x)*Hypergeometric2F1[-5/2, -1/2, 1
/2, (d*(a + b*x))/(-(b*c) + a*d)])/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(3*b^4*(b*c - a*d)*(a + b*x)^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(749\) vs. \(2(184)=368\).

Time = 1.70 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.38

method	result	size
default	\(\frac {\sqrt {d x +c}\, \left (105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} d^{3} x^{2}-150 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c \,d^{2} x^{2}+45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{4} c^{2} d \,x^{2}+12 b^{3} d^{2} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+210 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b \,d^{3} x -300 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c \,d^{2} x +90 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{2} d x -42 a \,b^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+54 b^{3} c d \,x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} d^{3}-150 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b c \,d^{2}+45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c^{2} d -280 a^{2} b \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+316 a \,b^{2} c d x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-48 b^{3} c^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-210 a^{3} d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+230 a^{2} b c d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-32 a \,b^{2} c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right )}{24 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \left (b x +a \right )^{\frac {3}{2}} b^{4}}\)	\(750\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.79 \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\left [\frac {15 \, {\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 7 \, a^{4} d^{2} + {\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 7 \, a^{3} b d^{2}\right )} x\right )} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (6 \, b^{3} d^{2} x^{3} - 16 \, a b^{2} c^{2} + 115 \, a^{2} b c d - 105 \, a^{3} d^{2} + 3 \, {\left (9 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} - 2 \, {\left (12 \, b^{3} c^{2} - 79 \, a b^{2} c d + 70 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {15 \, {\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 7 \, a^{4} d^{2} + {\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 7 \, a^{3} b d^{2}\right )} x\right )} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) - 2 \, {\left (6 \, b^{3} d^{2} x^{3} - 16 \, a b^{2} c^{2} + 115 \, a^{2} b c d - 105 \, a^{3} d^{2} + 3 \, {\left (9 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} - 2 \, {\left (12 \, b^{3} c^{2} - 79 \, a b^{2} c d + 70 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {x \left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x*(d*x+c)**(5/2)/(b*x+a)**(5/2),x)

[Out]

Integral(x*(c + d*x)**(5/2)/(a + b*x)**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x*(d*x+c)^(5/2)/(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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 840 vs. \(2 (184) = 368\).

Time = 0.52 (sec) , antiderivative size = 840, normalized size of antiderivative = 3.78 \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {1}{4} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{6}} + \frac {9 \, b^{12} c d^{3} {\left | b \right |} - 13 \, a b^{11} d^{4} {\left | b \right |}}{b^{17} d^{2}}\right )} - \frac {5 \, {\left (3 \, \sqrt {b d} b^{2} c^{2} {\left | b \right |} - 10 \, \sqrt {b d} a b c d {\left | b \right |} + 7 \, \sqrt {b d} a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\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 )}{8 \, b^{6}} - \frac {4 \, {\left (3 \, \sqrt {b d} b^{7} c^{5} {\left | b \right |} - 22 \, \sqrt {b d} a b^{6} c^{4} d {\left | b \right |} + 58 \, \sqrt {b d} a^{2} b^{5} c^{3} d^{2} {\left | b \right |} - 72 \, \sqrt {b d} a^{3} b^{4} c^{2} d^{3} {\left | b \right |} + 43 \, \sqrt {b d} a^{4} b^{3} c d^{4} {\left | b \right |} - 10 \, \sqrt {b d} a^{5} b^{2} d^{5} {\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} b^{5} c^{4} {\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 b^{4} 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^{3} c^{2} d^{2} {\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^{3} b^{2} c d^{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 )}^{2} a^{4} b d^{4} {\left | b \right |} + 3 \, \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^{3} 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^{2} c^{2} d {\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^{2} b c d^{2} {\left | b \right |} - 12 \, \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} d^{3} {\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} b^{5}} \]

[In]

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

[Out]

1/4*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*d^2*abs(b)/b^6 + (9*b^12*c*d^3*abs(b) - 13*
a*b^11*d^4*abs(b))/(b^17*d^2)) - 5/8*(3*sqrt(b*d)*b^2*c^2*abs(b) - 10*sqrt(b*d)*a*b*c*d*abs(b) + 7*sqrt(b*d)*a
^2*d^2*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/b^6 - 4/3*(3*sqrt(b*d)*b
^7*c^5*abs(b) - 22*sqrt(b*d)*a*b^6*c^4*d*abs(b) + 58*sqrt(b*d)*a^2*b^5*c^3*d^2*abs(b) - 72*sqrt(b*d)*a^3*b^4*c
^2*d^3*abs(b) + 43*sqrt(b*d)*a^4*b^3*c*d^4*abs(b) - 10*sqrt(b*d)*a^5*b^2*d^5*abs(b) - 6*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^5*c^4*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*b^4*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^3*c^2*d^2*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^3*b^2*c*d^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))^2*a^4*b*d^4*abs(b) + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^4*b^3*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^2*c^2*d
*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^2*b*c*d^2*abs(b) -
12*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*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)^3*b^5)

Mupad [F(-1)]

Timed out. \[ \int \frac {x (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {x\,{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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