3.1781 \(\int \frac{A+B x}{(d+e x) (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=302 \[ -\frac{e^2 (B d-A e)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{e^3 (a+b x) \log (a+b x) (B d-A e)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{e^3 (a+b x) (B d-A e) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{e (B d-A e)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{B d-A e}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

-((e^2*(B*d - A*e))/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (A*b - a*B)/(4*b*(b*d - a*e)*(a + b*x)^3*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (B*d - A*e)/(3*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e*
(B*d - A*e))/(2*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^3*(B*d - A*e)*(a + b*x)*Log[a + b*
x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^3*(B*d - A*e)*(a + b*x)*Log[d + e*x])/((b*d - a*e)^5*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.258353, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ -\frac{e^2 (B d-A e)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{e^3 (a+b x) \log (a+b x) (B d-A e)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{e^3 (a+b x) (B d-A e) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{e (B d-A e)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{B d-A e}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e^2*(B*d - A*e))/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (A*b - a*B)/(4*b*(b*d - a*e)*(a + b*x)^3*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (B*d - A*e)/(3*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e*
(B*d - A*e))/(2*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^3*(B*d - A*e)*(a + b*x)*Log[a + b*
x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^3*(B*d - A*e)*(a + b*x)*Log[d + e*x])/((b*d - a*e)^5*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{A+B x}{\left (a b+b^2 x\right )^5 (d+e x)} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac{A b-a B}{b^5 (b d-a e) (a+b x)^5}+\frac{B d-A e}{b^4 (b d-a e)^2 (a+b x)^4}+\frac{e (-B d+A e)}{b^4 (b d-a e)^3 (a+b x)^3}-\frac{e^2 (-B d+A e)}{b^4 (b d-a e)^4 (a+b x)^2}+\frac{e^3 (-B d+A e)}{b^4 (b d-a e)^5 (a+b x)}-\frac{e^4 (-B d+A e)}{b^5 (b d-a e)^5 (d+e x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{e^2 (B d-A e)}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{B d-A e}{3 (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e (B d-A e)}{2 (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e^3 (B d-A e) (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 (B d-A e) (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.158602, size = 182, normalized size = 0.6 \[ \frac{12 e^2 (a+b x)^2 (b d-a e) (A e-B d)+12 e^3 (a+b x)^3 \log (a+b x) (A e-B d)+12 e^3 (a+b x)^3 (B d-A e) \log (d+e x)+\frac{3 (a B-A b) (b d-a e)^4}{b (a+b x)}-6 e (a+b x) (b d-a e)^2 (A e-B d)+4 (b d-a e)^3 (A e-B d)}{12 \left ((a+b x)^2\right )^{3/2} (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(b*d - a*e)^3*(-(B*d) + A*e) + (3*(-(A*b) + a*B)*(b*d - a*e)^4)/(b*(a + b*x)) - 6*e*(b*d - a*e)^2*(-(B*d) +
A*e)*(a + b*x) + 12*e^2*(b*d - a*e)*(-(B*d) + A*e)*(a + b*x)^2 + 12*e^3*(-(B*d) + A*e)*(a + b*x)^3*Log[a + b*
x] + 12*e^3*(B*d - A*e)*(a + b*x)^3*Log[d + e*x])/(12*(b*d - a*e)^5*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.02, size = 776, normalized size = 2.6 \begin{align*}{\frac{ \left ( -48\,A{x}^{2}a{b}^{4}d{e}^{3}+3\,A{b}^{5}{d}^{4}-3\,B{a}^{5}{e}^{4}+48\,B\ln \left ( bx+a \right ){x}^{3}a{b}^{4}d{e}^{3}-48\,B\ln \left ( ex+d \right ){x}^{3}a{b}^{4}d{e}^{3}-72\,B\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{3}d{e}^{3}-48\,B\ln \left ( ex+d \right ) x{a}^{3}{b}^{2}d{e}^{3}-48\,A{a}^{3}{b}^{2}d{e}^{3}-10\,B{a}^{4}bd{e}^{3}+18\,B{a}^{3}{b}^{2}{d}^{2}{e}^{2}-72\,A\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}{e}^{4}-12\,B{x}^{3}a{b}^{4}d{e}^{3}+24\,Axa{b}^{4}{d}^{2}{e}^{2}-24\,Bxa{b}^{4}{d}^{3}e+48\,A\ln \left ( ex+d \right ){x}^{3}a{b}^{4}{e}^{4}+72\,A\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{3}{e}^{4}+48\,A\ln \left ( ex+d \right ) x{a}^{3}{b}^{2}{e}^{4}-12\,B\ln \left ( ex+d \right ){a}^{4}bd{e}^{3}-12\,B\ln \left ( ex+d \right ){x}^{4}{b}^{5}d{e}^{3}-42\,B{x}^{2}{a}^{2}{b}^{3}d{e}^{3}+48\,B{x}^{2}a{b}^{4}{d}^{2}{e}^{2}-72\,Ax{a}^{2}{b}^{3}d{e}^{3}-52\,Bx{a}^{3}{b}^{2}d{e}^{3}+72\,Bx{a}^{2}{b}^{3}{d}^{2}{e}^{2}+12\,B\ln \left ( bx+a \right ){a}^{4}bd{e}^{3}-48\,A\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}{e}^{4}-48\,A\ln \left ( bx+a \right ){x}^{3}a{b}^{4}{e}^{4}+12\,B\ln \left ( bx+a \right ){x}^{4}{b}^{5}d{e}^{3}+4\,Bx{b}^{5}{d}^{4}+Ba{b}^{4}{d}^{4}+25\,A{a}^{4}b{e}^{4}+12\,A\ln \left ( ex+d \right ){x}^{4}{b}^{5}{e}^{4}+12\,A\ln \left ( ex+d \right ){a}^{4}b{e}^{4}+42\,A{x}^{2}{a}^{2}{b}^{3}{e}^{4}-12\,A\ln \left ( bx+a \right ){a}^{4}b{e}^{4}+12\,A{x}^{3}a{b}^{4}{e}^{4}-12\,A{x}^{3}{b}^{5}d{e}^{3}+12\,B{x}^{3}{b}^{5}{d}^{2}{e}^{2}+52\,Ax{a}^{3}{b}^{2}{e}^{4}-4\,Ax{b}^{5}{d}^{3}e-12\,A\ln \left ( bx+a \right ){x}^{4}{b}^{5}{e}^{4}+6\,A{x}^{2}{b}^{5}{d}^{2}{e}^{2}-6\,B{x}^{2}{b}^{5}{d}^{3}e-16\,Aa{b}^{4}{d}^{3}e-6\,{b}^{3}B{a}^{2}{d}^{3}e+36\,A{a}^{2}{b}^{3}{d}^{2}{e}^{2}+48\,B\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}d{e}^{3}+72\,B\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}d{e}^{3} \right ) \left ( bx+a \right ) }{12\, \left ( ae-bd \right ) ^{5}b} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/12*(-48*A*x^2*a*b^4*d*e^3+3*A*b^5*d^4-3*B*a^5*e^4+48*B*ln(b*x+a)*x^3*a*b^4*d*e^3-48*B*ln(e*x+d)*x^3*a*b^4*d*
e^3-72*B*ln(e*x+d)*x^2*a^2*b^3*d*e^3-48*B*ln(e*x+d)*x*a^3*b^2*d*e^3-48*A*a^3*b^2*d*e^3-10*B*a^4*b*d*e^3+18*B*a
^3*b^2*d^2*e^2-72*A*ln(b*x+a)*x^2*a^2*b^3*e^4-12*B*x^3*a*b^4*d*e^3+24*A*x*a*b^4*d^2*e^2-24*B*x*a*b^4*d^3*e+48*
A*ln(e*x+d)*x^3*a*b^4*e^4+72*A*ln(e*x+d)*x^2*a^2*b^3*e^4+48*A*ln(e*x+d)*x*a^3*b^2*e^4-12*B*ln(e*x+d)*a^4*b*d*e
^3-12*B*ln(e*x+d)*x^4*b^5*d*e^3-42*B*x^2*a^2*b^3*d*e^3+48*B*x^2*a*b^4*d^2*e^2-72*A*x*a^2*b^3*d*e^3-52*B*x*a^3*
b^2*d*e^3+72*B*x*a^2*b^3*d^2*e^2+12*B*ln(b*x+a)*a^4*b*d*e^3-48*A*ln(b*x+a)*x*a^3*b^2*e^4-48*A*ln(b*x+a)*x^3*a*
b^4*e^4+12*B*ln(b*x+a)*x^4*b^5*d*e^3+4*B*x*b^5*d^4+B*a*b^4*d^4+25*A*a^4*b*e^4+12*A*ln(e*x+d)*x^4*b^5*e^4+12*A*
ln(e*x+d)*a^4*b*e^4+42*A*x^2*a^2*b^3*e^4-12*A*ln(b*x+a)*a^4*b*e^4+12*A*x^3*a*b^4*e^4-12*A*x^3*b^5*d*e^3+12*B*x
^3*b^5*d^2*e^2+52*A*x*a^3*b^2*e^4-4*A*x*b^5*d^3*e-12*A*ln(b*x+a)*x^4*b^5*e^4+6*A*x^2*b^5*d^2*e^2-6*B*x^2*b^5*d
^3*e-16*A*a*b^4*d^3*e-6*b^3*B*a^2*d^3*e+36*A*a^2*b^3*d^2*e^2+48*B*ln(b*x+a)*x*a^3*b^2*d*e^3+72*B*ln(b*x+a)*x^2
*a^2*b^3*d*e^3)*(b*x+a)/b/(a*e-b*d)^5/((b*x+a)^2)^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.43315, size = 1932, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*((B*a*b^4 + 3*A*b^5)*d^4 - 2*(3*B*a^2*b^3 + 8*A*a*b^4)*d^3*e + 18*(B*a^3*b^2 + 2*A*a^2*b^3)*d^2*e^2 - 2*
(5*B*a^4*b + 24*A*a^3*b^2)*d*e^3 - (3*B*a^5 - 25*A*a^4*b)*e^4 + 12*(B*b^5*d^2*e^2 + A*a*b^4*e^4 - (B*a*b^4 + A
*b^5)*d*e^3)*x^3 - 6*(B*b^5*d^3*e - 7*A*a^2*b^3*e^4 - (8*B*a*b^4 + A*b^5)*d^2*e^2 + (7*B*a^2*b^3 + 8*A*a*b^4)*
d*e^3)*x^2 + 4*(B*b^5*d^4 + 13*A*a^3*b^2*e^4 - (6*B*a*b^4 + A*b^5)*d^3*e + 6*(3*B*a^2*b^3 + A*a*b^4)*d^2*e^2 -
(13*B*a^3*b^2 + 18*A*a^2*b^3)*d*e^3)*x + 12*(B*a^4*b*d*e^3 - A*a^4*b*e^4 + (B*b^5*d*e^3 - A*b^5*e^4)*x^4 + 4*
(B*a*b^4*d*e^3 - A*a*b^4*e^4)*x^3 + 6*(B*a^2*b^3*d*e^3 - A*a^2*b^3*e^4)*x^2 + 4*(B*a^3*b^2*d*e^3 - A*a^3*b^2*e
^4)*x)*log(b*x + a) - 12*(B*a^4*b*d*e^3 - A*a^4*b*e^4 + (B*b^5*d*e^3 - A*b^5*e^4)*x^4 + 4*(B*a*b^4*d*e^3 - A*a
*b^4*e^4)*x^3 + 6*(B*a^2*b^3*d*e^3 - A*a^2*b^3*e^4)*x^2 + 4*(B*a^3*b^2*d*e^3 - A*a^3*b^2*e^4)*x)*log(e*x + d))
/(a^4*b^6*d^5 - 5*a^5*b^5*d^4*e + 10*a^6*b^4*d^3*e^2 - 10*a^7*b^3*d^2*e^3 + 5*a^8*b^2*d*e^4 - a^9*b*e^5 + (b^1
0*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5*b^5*e^5)*x^4 + 4*(a*b^
9*d^5 - 5*a^2*b^8*d^4*e + 10*a^3*b^7*d^3*e^2 - 10*a^4*b^6*d^2*e^3 + 5*a^5*b^5*d*e^4 - a^6*b^4*e^5)*x^3 + 6*(a^
2*b^8*d^5 - 5*a^3*b^7*d^4*e + 10*a^4*b^6*d^3*e^2 - 10*a^5*b^5*d^2*e^3 + 5*a^6*b^4*d*e^4 - a^7*b^3*e^5)*x^2 + 4
*(a^3*b^7*d^5 - 5*a^4*b^6*d^4*e + 10*a^5*b^5*d^3*e^2 - 10*a^6*b^4*d^2*e^3 + 5*a^7*b^3*d*e^4 - a^8*b^2*e^5)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x