3.8.39 \(\int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=108 \[ \frac {3 \sqrt {c} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}-\frac {3 \sqrt {c+d x} (b c-a d)}{a^2 \sqrt {a+b x}}-\frac {(c+d x)^{3/2}}{a x \sqrt {a+b x}} \]

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Rubi [A]  time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {94, 93, 208} \begin {gather*} -\frac {3 \sqrt {c+d x} (b c-a d)}{a^2 \sqrt {a+b x}}+\frac {3 \sqrt {c} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}-\frac {(c+d x)^{3/2}}{a x \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 93

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 94

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[p, 1] &&  !SumSimplerQ[m, 1])

Rule 208

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

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 91, normalized size = 0.84 \begin {gather*} \frac {3 \sqrt {c} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}+\frac {\sqrt {c+d x} (-a c+2 a d x-3 b c x)}{a^2 x \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 1.19, size = 238, normalized size = 2.20 \begin {gather*} \frac {3 \sqrt {\frac {b}{d}} \left (b c^{3/2} \sqrt {d}-a \sqrt {c} d^{3/2}\right ) \tanh ^{-1}\left (\frac {d \sqrt {\frac {b}{d}} \sqrt {c+d x} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}-b (c+d x)+b c}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )}{a^{5/2} \sqrt {b}}-\frac {\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}} \left (3 a c d^2 \sqrt {c+d x}-2 a d^2 (c+d x)^{3/2}-3 b c^2 d \sqrt {c+d x}+3 b c d (c+d x)^{3/2}\right )}{a^2 d x (a d+b (c+d x)-b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.75, size = 341, normalized size = 3.16 \begin {gather*} \left [-\frac {3 \, {\left ({\left (b^{2} c - a b d\right )} x^{2} + {\left (a b c - a^{2} d\right )} x\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 (a c + {\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (a^{2} b x^{2} + a^{3} x\right )}}, -\frac {3 \, {\left ({\left (b^{2} c - a b d\right )} x^{2} + {\left (a b c - a^{2} d\right )} x\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 (a c + {\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a^{2} b x^{2} + a^{3} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(3*((b^2*c - a*b*d)*x^2 + (a*b*c - a^2*d)*x)*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*(a*c + (3*b*c - 2*a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b*x^2 + a^3*x), -1/2*(3*((b^2*c - a*b*d)*x^2 + (
a*b*c - a^2*d)*x)*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*(a*c + (3*b*c - 2*a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b*x^2 + a^3*
x)]

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giac [B]  time = 6.41, size = 904, normalized size = 8.37 \begin {gather*} \frac {3 \, {\left (\sqrt {b d} b c^{2} {\left | b \right |} - \sqrt {b d} a c d {\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 )}{\sqrt {-a b c d} a^{2} b} - \frac {2 \, {\left (3 \, \sqrt {b d} b^{6} c^{4} {\left | b \right |} - 11 \, \sqrt {b d} a b^{5} c^{3} d {\left | b \right |} + 15 \, \sqrt {b d} a^{2} b^{4} c^{2} d^{2} {\left | b \right |} - 9 \, \sqrt {b d} a^{3} b^{3} c d^{3} {\left | b \right |} + 2 \, \sqrt {b d} a^{4} b^{2} d^{4} {\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^{4} c^{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 b^{3} c^{2} d {\left | b \right |} + 4 \, \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 d^{2} {\left | b \right |} - 4 \, \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 d^{3} {\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^{2} c^{2} {\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} a b c d {\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 )}^{4} a^{2} d^{2} {\left | b \right |}\right )}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3} - 3 \, {\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^{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} a b^{3} c 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} d^{2} + 3 \, {\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 + {\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 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}\right )} a^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(sqrt(b*d)*b*c^2*abs(b) - sqrt(b*d)*a*c*d*abs(b))*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^2*b) - 2*(3*sqrt(b*d)*b^6*c^4*abs(
b) - 11*sqrt(b*d)*a*b^5*c^3*d*abs(b) + 15*sqrt(b*d)*a^2*b^4*c^2*d^2*abs(b) - 9*sqrt(b*d)*a^3*b^3*c*d^3*abs(b)
+ 2*sqrt(b*d)*a^4*b^2*d^4*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*b^4*c^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*b^3*c^2*d*
abs(b) + 4*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*d^2*abs(b) -
4*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*d^3*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^2*c^2*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*a*b*c*d*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))^4*a^2*d^2*abs(b))/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3 -
3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^4*c^2 + 2*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^3*c*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*d^2 + 3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^2*c + (sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*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)

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maple [B]  time = 0.02, size = 298, normalized size = 2.76 \begin {gather*} -\frac {\sqrt {d x +c}\, \left (3 a b c d \,x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-3 b^{2} c^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 a^{2} c d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-3 a b \,c^{2} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a d x +6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b c x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a c \right )}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, \sqrt {b x +a}\, a^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(d*x+c)^(1/2)*(3*a*b*c*d*x^2*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)-3*b^2*c^2*x^
2*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)+3*a^2*c*d*x*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2))/x)-3*a*b*c^2*x*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)-4*
((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*d*x+6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*b*c*x+2*((b*x+a)*(d*x+c))^(1/2
)*(a*c)^(1/2)*a*c)/a^2/((b*x+a)*(d*x+c))^(1/2)/x/(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)^(3/2)/x^2/(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 details)Is a*d-b*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{3/2}}{x^2\,{\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)^(3/2)/(x^2*(a + b*x)^(3/2)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{2} \left (a + b x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(3/2)/x**2/(b*x+a)**(3/2),x)

[Out]

Integral((c + d*x)**(3/2)/(x**2*(a + b*x)**(3/2)), x)

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