3.18.41 \(\int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=146 \[ -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^3 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{6 e^3 (a+b x) (d+e x)^6} \]

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Rubi [A]  time = 0.08, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^3 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{6 e^3 (a+b x) (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^7,x]

[Out]

-((b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^3*(a + b*x)*(d + e*x)^6) + (2*b*(b*d - a*e)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(5*e^3*(a + b*x)*(d + e*x)^5) - (b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3*(a + b*x)*(d + e*x
)^4)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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]

Rubi steps

\begin {align*} \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )}{(d+e x)^7} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^2}{(d+e x)^7} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^7}-\frac {2 b (b d-a e)}{e^2 (d+e x)^6}+\frac {b^2}{e^2 (d+e x)^5}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x) (d+e x)^6}+\frac {2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 73, normalized size = 0.50 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )}{60 e^3 (a+b x) (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^7,x]

[Out]

-1/60*(Sqrt[(a + b*x)^2]*(10*a^2*e^2 + 4*a*b*e*(d + 6*e*x) + b^2*(d^2 + 6*d*e*x + 15*e^2*x^2)))/(e^3*(a + b*x)
*(d + e*x)^6)

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IntegrateAlgebraic [F]  time = 180.02, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^7,x]

[Out]

$Aborted

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fricas [A]  time = 0.40, size = 120, normalized size = 0.82 \begin {gather*} -\frac {15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \, {\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

-1/60*(15*b^2*e^2*x^2 + b^2*d^2 + 4*a*b*d*e + 10*a^2*e^2 + 6*(b^2*d*e + 4*a*b*e^2)*x)/(e^9*x^6 + 6*d*e^8*x^5 +
15*d^2*e^7*x^4 + 20*d^3*e^6*x^3 + 15*d^4*e^5*x^2 + 6*d^5*e^4*x + d^6*e^3)

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giac [A]  time = 0.16, size = 96, normalized size = 0.66 \begin {gather*} -\frac {{\left (15 \, b^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{2} d x e \mathrm {sgn}\left (b x + a\right ) + b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + 24 \, a b x e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b d e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/60*(15*b^2*x^2*e^2*sgn(b*x + a) + 6*b^2*d*x*e*sgn(b*x + a) + b^2*d^2*sgn(b*x + a) + 24*a*b*x*e^2*sgn(b*x +
a) + 4*a*b*d*e*sgn(b*x + a) + 10*a^2*e^2*sgn(b*x + a))*e^(-3)/(x*e + d)^6

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maple [A]  time = 0.05, size = 78, normalized size = 0.53 \begin {gather*} -\frac {\left (15 b^{2} e^{2} x^{2}+24 a b \,e^{2} x +6 b^{2} d e x +10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{60 \left (e x +d \right )^{6} \left (b x +a \right ) e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^7,x)

[Out]

-1/60/e^3*(15*b^2*e^2*x^2+24*a*b*e^2*x+6*b^2*d*e*x+10*a^2*e^2+4*a*b*d*e+b^2*d^2)*((b*x+a)^2)^(1/2)/(e*x+d)^6/(
b*x+a)

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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)^2)^(1/2)/(e*x+d)^7,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 zero or nonzero?

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mupad [B]  time = 2.12, size = 77, normalized size = 0.53 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (10\,a^2\,e^2+4\,a\,b\,d\,e+24\,a\,b\,e^2\,x+b^2\,d^2+6\,b^2\,d\,e\,x+15\,b^2\,e^2\,x^2\right )}{60\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((a + b*x)^2)^(1/2)*(a + b*x))/(d + e*x)^7,x)

[Out]

-(((a + b*x)^2)^(1/2)*(10*a^2*e^2 + b^2*d^2 + 15*b^2*e^2*x^2 + 24*a*b*e^2*x + 6*b^2*d*e*x + 4*a*b*d*e))/(60*e^
3*(a + b*x)*(d + e*x)^6)

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sympy [A]  time = 1.22, size = 128, normalized size = 0.88 \begin {gather*} \frac {- 10 a^{2} e^{2} - 4 a b d e - b^{2} d^{2} - 15 b^{2} e^{2} x^{2} + x \left (- 24 a b e^{2} - 6 b^{2} d e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d)**7,x)

[Out]

(-10*a**2*e**2 - 4*a*b*d*e - b**2*d**2 - 15*b**2*e**2*x**2 + x*(-24*a*b*e**2 - 6*b**2*d*e))/(60*d**6*e**3 + 36
0*d**5*e**4*x + 900*d**4*e**5*x**2 + 1200*d**3*e**6*x**3 + 900*d**2*e**7*x**4 + 360*d*e**8*x**5 + 60*e**9*x**6
)

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