3.6.89 \(\int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^5} \, dx\)

Optimal. Leaf size=114 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^3 (a+b x)}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]

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Rubi [A]  time = 0.04, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 76} \begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^3 (a+b x)}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(a + b*x)) - ((A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a
+ b*x)) - (b*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x))

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

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

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Mathematica [A]  time = 0.01, size = 47, normalized size = 0.41 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (3 a A+4 a B x+4 A b x+6 b B x^2\right )}{12 x^4 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(Sqrt[(a + b*x)^2]*(3*a*A + 4*A*b*x + 4*a*B*x + 6*b*B*x^2))/(x^4*(a + b*x))

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IntegrateAlgebraic [B]  time = 24.77, size = 911, normalized size = 7.99 \begin {gather*} \frac {2 b^3 (a+b x)^3 (a+2 b x)^{35} \left (6 b B x^2+4 A b x+4 a B x+3 a A\right )}{3 \sqrt {a^2+2 b x a+b^2 x^2} \left (-137438953472 x^{37} b^{40}-2680059592704 a x^{36} b^{39}-25391846653952 a^2 x^{35} b^{38}-155735514152960 a^3 x^{34} b^{37}-695097507184640 a^4 x^{33} b^{36}-2406264017518592 a^5 x^{32} b^{35}-6724046179794944 a^6 x^{31} b^{34}-15586101309734912 a^7 x^{30} b^{33}-30562203446804480 a^8 x^{29} b^{32}-51445338388561920 a^9 x^{28} b^{31}-75182398031003648 a^{10} x^{27} b^{30}-96232320779943936 a^{11} x^{26} b^{29}-108638277979865088 a^{12} x^{25} b^{28}-108767506700697600 a^{13} x^{24} b^{27}-96998462482022400 a^{14} x^{23} b^{26}-77314159112355840 a^{15} x^{22} b^{25}-55221240075386880 a^{16} x^{21} b^{24}-35409421758627840 a^{17} x^{20} b^{23}-20409249457766400 a^{18} x^{19} b^{22}-10580255087001600 a^{19} x^{18} b^{21}-4933278703288320 a^{20} x^{17} b^{20}-2067755685642240 a^{21} x^{16} b^{19}-778172761374720 a^{22} x^{15} b^{18}-262465791590400 a^{23} x^{14} b^{17}-79137868185600 a^{24} x^{13} b^{16}-21259428642816 a^{25} x^{12} b^{15}-5066401062912 a^{26} x^{11} b^{14}-1065250041856 a^{27} x^{10} b^{13}-196248391680 a^{28} x^9 b^{12}-31402117120 a^{29} x^8 b^{11}-4315565056 a^{30} x^7 b^{10}-501985792 a^{31} x^6 b^9-48464416 a^{32} x^5 b^8-3779440 a^{33} x^4 b^7-228760 a^{34} x^3 b^6-10084 a^{35} x^2 b^5-288 a^{36} x b^4-4 a^{37} b^3\right ) x^4+3 \sqrt {b^2} \left (137438953472 x^{38} b^{40}[phone] a x^{37} b^{39}[phone] a^2 x^{36} b^{38}[phone] a^3 x^{35} b^{37}[phone] a^4 x^{34} b^{36}+3101361524703232 a^5 x^{33} b^{35}+9130310197313536 a^6 x^{32} b^{34}+22310147489529856 a^7 x^{31} b^{33}+46148304756539392 a^8 x^{30} b^{32}+82007541835366400 a^9 x^{29} b^{31}+126627736419565568 a^{10} x^{28} b^{30}+171414718810947584 a^{11} x^{27} b^{29}+204870598759809024 a^{12} x^{26} b^{28}+217405784680562688 a^{13} x^{25} b^{27}+205765969182720000 a^{14} x^{24} b^{26}+174312621594378240 a^{15} x^{23} b^{25}+132535399187742720 a^{16} x^{22} b^{24}+90630661834014720 a^{17} x^{21} b^{23}+55818671216394240 a^{18} x^{20} b^{22}+30989504544768000 a^{19} x^{19} b^{21}+15513533790289920 a^{20} x^{18} b^{20}+7001034388930560 a^{21} x^{17} b^{19}+2845928447016960 a^{22} x^{16} b^{18}+1040638552965120 a^{23} x^{15} b^{17}[phone] a^{24} x^{14} b^{16}[phone] a^{25} x^{13} b^{15}[phone] a^{26} x^{12} b^{14}[phone] a^{27} x^{11} b^{13}[phone] a^{28} x^{10} b^{12}[phone] a^{29} x^9 b^{11}[phone] a^{30} x^8 b^{10}[phone] a^{31} x^7 b^9+550450208 a^{32} x^6 b^8+52243856 a^{33} x^5 b^7+4008200 a^{34} x^4 b^6+238844 a^{35} x^3 b^5+10372 a^{36} x^2 b^4+292 a^{37} x b^3+4 a^{38} b^2\right ) x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^5,x]

[Out]

(2*b^3*(a + b*x)^3*(a + 2*b*x)^35*(3*a*A + 4*A*b*x + 4*a*B*x + 6*b*B*x^2))/(3*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]*(-4*a^37*b^3 - 288*a^36*b^4*x - 10084*a^35*b^5*x^2 - 228760*a^34*b^6*x^3 - 3779440*a^33*b^7*x^4 - 48464416*a
^32*b^8*x^5 - 501985792*a^31*b^9*x^6 - 4315565056*a^30*b^10*x^7 - 31402117120*a^29*b^11*x^8 - 196248391680*a^2
8*b^12*x^9 - 1065250041856*a^27*b^13*x^10 - 5066401062912*a^26*b^14*x^11 - 21259428642816*a^25*b^15*x^12 - 791
37868185600*a^24*b^16*x^13 - 262465791590400*a^23*b^17*x^14 - 778172761374720*a^22*b^18*x^15 - 206775568564224
0*a^21*b^19*x^16 - 4933278703288320*a^20*b^20*x^17 - 10580255087001600*a^19*b^21*x^18 - 20409249457766400*a^18
*b^22*x^19 - 35409421758627840*a^17*b^23*x^20 - 55221240075386880*a^16*b^24*x^21 - 77314159112355840*a^15*b^25
*x^22 - 96998462482022400*a^14*b^26*x^23 - 108767506700697600*a^13*b^27*x^24 - 108638277979865088*a^12*b^28*x^
25 - 96232320779943936*a^11*b^29*x^26 - 75182398031003648*a^10*b^30*x^27 - 51445338388561920*a^9*b^31*x^28 - 3
0562203446804480*a^8*b^32*x^29 - 15586101309734912*a^7*b^33*x^30 - 6724046179794944*a^6*b^34*x^31 - 2406264017
518592*a^5*b^35*x^32 - 695097507184640*a^4*b^36*x^33 - 155735514152960*a^3*b^37*x^34 - 25391846653952*a^2*b^38
*x^35 - 2680059592704*a*b^39*x^36 - 137438953472*b^40*x^37) + 3*Sqrt[b^2]*x^4*(4*a^38*b^2 + 292*a^37*b^3*x + 1
0372*a^36*b^4*x^2 + 238844*a^35*b^5*x^3 + 4008200*a^34*b^6*x^4 + 52243856*a^33*b^7*x^5 + 550450208*a^32*b^8*x^
6 + 4817550848*a^31*b^9*x^7 + 35717682176*a^30*b^10*x^8 + 227650508800*a^29*b^11*x^9 + 1261498433536*a^28*b^12
*x^10 + 6131651104768*a^27*b^13*x^11 + 26325829705728*a^26*b^14*x^12 + 100397296828416*a^25*b^15*x^13 + 341603
659776000*a^24*b^16*x^14 + 1040638552965120*a^23*b^17*x^15 + 2845928447016960*a^22*b^18*x^16 + 700103438893056
0*a^21*b^19*x^17 + 15513533790289920*a^20*b^20*x^18 + 30989504544768000*a^19*b^21*x^19 + 55818671216394240*a^1
8*b^22*x^20 + 90630661834014720*a^17*b^23*x^21 + 132535399187742720*a^16*b^24*x^22 + 174312621594378240*a^15*b
^25*x^23 + 205765969182720000*a^14*b^26*x^24 + 217405784680562688*a^13*b^27*x^25 + 204870598759809024*a^12*b^2
8*x^26 + 171414718810947584*a^11*b^29*x^27 + 126627736419565568*a^10*b^30*x^28 + 82007541835366400*a^9*b^31*x^
29 + 46148304756539392*a^8*b^32*x^30 + 22310147489529856*a^7*b^33*x^31 + 9130310197313536*a^6*b^34*x^32 + 3101
361524703232*a^5*b^35*x^33 + 850833021337600*a^4*b^36*x^34 + 181127360806912*a^3*b^37*x^35 + 28071906246656*a^
2*b^38*x^36 + 2817498546176*a*b^39*x^37 + 137438953472*b^40*x^38))

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fricas [A]  time = 0.40, size = 27, normalized size = 0.24 \begin {gather*} -\frac {6 \, B b x^{2} + 3 \, A a + 4 \, {\left (B a + A b\right )} x}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/x^5,x, algorithm="fricas")

[Out]

-1/12*(6*B*b*x^2 + 3*A*a + 4*(B*a + A*b)*x)/x^4

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giac [A]  time = 0.16, size = 77, normalized size = 0.68 \begin {gather*} \frac {{\left (2 \, B a b^{3} - A b^{4}\right )} \mathrm {sgn}\left (b x + a\right )}{12 \, a^{3}} - \frac {6 \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, B a x \mathrm {sgn}\left (b x + a\right ) + 4 \, A b x \mathrm {sgn}\left (b x + a\right ) + 3 \, A a \mathrm {sgn}\left (b x + a\right )}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/x^5,x, algorithm="giac")

[Out]

1/12*(2*B*a*b^3 - A*b^4)*sgn(b*x + a)/a^3 - 1/12*(6*B*b*x^2*sgn(b*x + a) + 4*B*a*x*sgn(b*x + a) + 4*A*b*x*sgn(
b*x + a) + 3*A*a*sgn(b*x + a))/x^4

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maple [A]  time = 0.06, size = 44, normalized size = 0.39 \begin {gather*} -\frac {\left (6 B b \,x^{2}+4 A b x +4 B a x +3 A a \right ) \sqrt {\left (b x +a \right )^{2}}}{12 \left (b x +a \right ) x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*((b*x+a)^2)^(1/2)/x^5,x)

[Out]

-1/12*(6*B*b*x^2+4*A*b*x+4*B*a*x+3*A*a)*((b*x+a)^2)^(1/2)/x^4/(b*x+a)

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maxima [B]  time = 0.54, size = 255, normalized size = 2.24 \begin {gather*} -\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{3}}{2 \, a^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{4}}{2 \, a^{4}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{2}}{2 \, a^{2} x} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{3}}{2 \, a^{3} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b}{2 \, a^{3} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{2}}{2 \, a^{4} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B}{3 \, a^{2} x^{3}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b}{12 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A}{4 \, a^{2} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/x^5,x, algorithm="maxima")

[Out]

-1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*b^3/a^3 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*b^4/a^4 - 1/2*sqrt(b^2*x^2
+ 2*a*b*x + a^2)*B*b^2/(a^2*x) + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*b^3/(a^3*x) + 1/2*(b^2*x^2 + 2*a*b*x + a^
2)^(3/2)*B*b/(a^3*x^2) - 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^2/(a^4*x^2) - 1/3*(b^2*x^2 + 2*a*b*x + a^2)^(
3/2)*B/(a^2*x^3) + 5/12*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b/(a^3*x^3) - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A/
(a^2*x^4)

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mupad [B]  time = 1.13, size = 43, normalized size = 0.38 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (3\,A\,a+4\,A\,b\,x+4\,B\,a\,x+6\,B\,b\,x^2\right )}{12\,x^4\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((a + b*x)^2)^(1/2)*(A + B*x))/x^5,x)

[Out]

-(((a + b*x)^2)^(1/2)*(3*A*a + 4*A*b*x + 4*B*a*x + 6*B*b*x^2))/(12*x^4*(a + b*x))

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sympy [A]  time = 0.37, size = 31, normalized size = 0.27 \begin {gather*} \frac {- 3 A a - 6 B b x^{2} + x \left (- 4 A b - 4 B a\right )}{12 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**5,x)

[Out]

(-3*A*a - 6*B*b*x**2 + x*(-4*A*b - 4*B*a))/(12*x**4)

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