3.17.8 \(\int \frac {A+B x}{(a+b x) (d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=88 \[ -\frac {2 (B d-A e)}{e \sqrt {d+e x} (b d-a e)}-\frac {2 (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}} \]

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Rubi [A]  time = 0.06, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {78, 63, 208} \begin {gather*} -\frac {2 (B d-A e)}{e \sqrt {d+e x} (b d-a e)}-\frac {2 (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(B*d - A*e))/(e*(b*d - a*e)*Sqrt[d + e*x]) - (2*(A*b - a*B)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e
]])/(Sqrt[b]*(b*d - a*e)^(3/2))

Rule 63

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 78

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] || LtQ
[p, n]))))

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

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Mathematica [A]  time = 0.32, size = 88, normalized size = 1.00 \begin {gather*} \frac {2 B d-2 A e}{e \sqrt {d+e x} (a e-b d)}+\frac {2 (a B-A b) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*d - 2*A*e)/(e*(-(b*d) + a*e)*Sqrt[d + e*x]) + (2*(-(A*b) + a*B)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(Sqrt[b]*(b*d - a*e)^(3/2))

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IntegrateAlgebraic [A]  time = 0.18, size = 98, normalized size = 1.11 \begin {gather*} \frac {2 (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{\sqrt {b} (a e-b d)^{3/2}}-\frac {2 (A e-B d)}{e \sqrt {d+e x} (a e-b d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-(B*d) + A*e))/(e*(-(b*d) + a*e)*Sqrt[d + e*x]) + (2*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[
d + e*x])/(b*d - a*e)])/(Sqrt[b]*(-(b*d) + a*e)^(3/2))

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fricas [B]  time = 1.58, size = 363, normalized size = 4.12 \begin {gather*} \left [\frac {{\left ({\left (B a - A b\right )} e^{2} x + {\left (B a - A b\right )} d e\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (B b^{2} d^{2} + A a b e^{2} - {\left (B a b + A b^{2}\right )} d e\right )} \sqrt {e x + d}}{b^{3} d^{3} e - 2 \, a b^{2} d^{2} e^{2} + a^{2} b d e^{3} + {\left (b^{3} d^{2} e^{2} - 2 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x}, -\frac {2 \, {\left ({\left ({\left (B a - A b\right )} e^{2} x + {\left (B a - A b\right )} d e\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left (B b^{2} d^{2} + A a b e^{2} - {\left (B a b + A b^{2}\right )} d e\right )} \sqrt {e x + d}\right )}}{b^{3} d^{3} e - 2 \, a b^{2} d^{2} e^{2} + a^{2} b d e^{3} + {\left (b^{3} d^{2} e^{2} - 2 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(((B*a - A*b)*e^2*x + (B*a - A*b)*d*e)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e + 2*sqrt(b^2*d - a*b*e)*s
qrt(e*x + d))/(b*x + a)) - 2*(B*b^2*d^2 + A*a*b*e^2 - (B*a*b + A*b^2)*d*e)*sqrt(e*x + d))/(b^3*d^3*e - 2*a*b^2
*d^2*e^2 + a^2*b*d*e^3 + (b^3*d^2*e^2 - 2*a*b^2*d*e^3 + a^2*b*e^4)*x), -2*(((B*a - A*b)*e^2*x + (B*a - A*b)*d*
e)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (B*b^2*d^2 + A*a*b*e^2 - (B
*a*b + A*b^2)*d*e)*sqrt(e*x + d))/(b^3*d^3*e - 2*a*b^2*d^2*e^2 + a^2*b*d*e^3 + (b^3*d^2*e^2 - 2*a*b^2*d*e^3 +
a^2*b*e^4)*x)]

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giac [A]  time = 1.25, size = 93, normalized size = 1.06 \begin {gather*} -\frac {2 \, {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} {\left (b d - a e\right )}} - \frac {2 \, {\left (B d - A e\right )}}{{\left (b d e - a e^{2}\right )} \sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(B*a - A*b)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*(b*d - a*e)) - 2*(B*d - A*e)
/((b*d*e - a*e^2)*sqrt(x*e + d))

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maple [A]  time = 0.01, size = 142, normalized size = 1.61 \begin {gather*} -\frac {2 A b \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}+\frac {2 B a \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}-\frac {2 A}{\left (a e -b d \right ) \sqrt {e x +d}}+\frac {2 B d}{\left (a e -b d \right ) \sqrt {e x +d}\, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)/(e*x+d)^(3/2),x)

[Out]

-2/(a*e-b*d)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A*b+2/(a*e-b*d)/((a*e-b*d)*b)^(1/
2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*a-2/(a*e-b*d)/(e*x+d)^(1/2)*A+2/e/(a*e-b*d)/(e*x+d)^(1/2)*B*d

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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)/(b*x+a)/(e*x+d)^(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*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d positive or negative?

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mupad [B]  time = 0.11, size = 96, normalized size = 1.09 \begin {gather*} -\frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {b}\,\left (A\,b-B\,a\right )\,\sqrt {d+e\,x}}{\left (2\,A\,b-2\,B\,a\right )\,\sqrt {a\,e-b\,d}}\right )\,\left (A\,b-B\,a\right )}{\sqrt {b}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {2\,\left (A\,e-B\,d\right )}{e\,\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((a + b*x)*(d + e*x)^(3/2)),x)

[Out]

- (2*atan((2*b^(1/2)*(A*b - B*a)*(d + e*x)^(1/2))/((2*A*b - 2*B*a)*(a*e - b*d)^(1/2)))*(A*b - B*a))/(b^(1/2)*(
a*e - b*d)^(3/2)) - (2*(A*e - B*d))/(e*(a*e - b*d)*(d + e*x)^(1/2))

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sympy [A]  time = 30.16, size = 76, normalized size = 0.86 \begin {gather*} \frac {2 \left (- A e + B d\right )}{e \sqrt {d + e x} \left (a e - b d\right )} + \frac {2 \left (- A b + B a\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b \sqrt {\frac {a e - b d}{b}} \left (a e - b d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)/(e*x+d)**(3/2),x)

[Out]

2*(-A*e + B*d)/(e*sqrt(d + e*x)*(a*e - b*d)) + 2*(-A*b + B*a)*atan(sqrt(d + e*x)/sqrt((a*e - b*d)/b))/(b*sqrt(
(a*e - b*d)/b)*(a*e - b*d))

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