3.3.41 \(\int \frac {A+B x}{x^{7/2} (b x+c x^2)^{3/2}} \, dx\) [241]

Optimal. Leaf size=216 \[ -\frac {A}{4 b x^{7/2} \sqrt {b x+c x^2}}-\frac {8 b B-9 A c}{24 b^2 x^{5/2} \sqrt {b x+c x^2}}+\frac {7 c (8 b B-9 A c)}{96 b^3 x^{3/2} \sqrt {b x+c x^2}}-\frac {35 c^2 (8 b B-9 A c)}{192 b^4 \sqrt {x} \sqrt {b x+c x^2}}-\frac {35 c^3 (8 b B-9 A c) \sqrt {x}}{64 b^5 \sqrt {b x+c x^2}}+\frac {35 c^3 (8 b B-9 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{64 b^{11/2}} \]

[Out]

35/64*c^3*(-9*A*c+8*B*b)*arctanh((c*x^2+b*x)^(1/2)/b^(1/2)/x^(1/2))/b^(11/2)-1/4*A/b/x^(7/2)/(c*x^2+b*x)^(1/2)
+1/24*(9*A*c-8*B*b)/b^2/x^(5/2)/(c*x^2+b*x)^(1/2)+7/96*c*(-9*A*c+8*B*b)/b^3/x^(3/2)/(c*x^2+b*x)^(1/2)-35/192*c
^2*(-9*A*c+8*B*b)/b^4/x^(1/2)/(c*x^2+b*x)^(1/2)-35/64*c^3*(-9*A*c+8*B*b)*x^(1/2)/b^5/(c*x^2+b*x)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {806, 686, 680, 674, 213} \begin {gather*} \frac {35 c^3 (8 b B-9 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{64 b^{11/2}}-\frac {35 c^3 \sqrt {x} (8 b B-9 A c)}{64 b^5 \sqrt {b x+c x^2}}-\frac {35 c^2 (8 b B-9 A c)}{192 b^4 \sqrt {x} \sqrt {b x+c x^2}}+\frac {7 c (8 b B-9 A c)}{96 b^3 x^{3/2} \sqrt {b x+c x^2}}-\frac {8 b B-9 A c}{24 b^2 x^{5/2} \sqrt {b x+c x^2}}-\frac {A}{4 b x^{7/2} \sqrt {b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(7/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

-1/4*A/(b*x^(7/2)*Sqrt[b*x + c*x^2]) - (8*b*B - 9*A*c)/(24*b^2*x^(5/2)*Sqrt[b*x + c*x^2]) + (7*c*(8*b*B - 9*A*
c))/(96*b^3*x^(3/2)*Sqrt[b*x + c*x^2]) - (35*c^2*(8*b*B - 9*A*c))/(192*b^4*Sqrt[x]*Sqrt[b*x + c*x^2]) - (35*c^
3*(8*b*B - 9*A*c)*Sqrt[x])/(64*b^5*Sqrt[b*x + c*x^2]) + (35*c^3*(8*b*B - 9*A*c)*ArcTanh[Sqrt[b*x + c*x^2]/(Sqr
t[b]*Sqrt[x])])/(64*b^(11/2))

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 680

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*c*d - b*e)*(d + e
*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(
b^2 - 4*a*c))), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ[0, m, 1] && IntegerQ[2*p]

Rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a
+ b*x + c*x^2)^(p + 1)/((m + p + 1)*(2*c*d - b*e))), x] + Dist[c*((m + 2*p + 2)/((m + p + 1)*(2*c*d - b*e))),
Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rubi steps

\begin {align*} \int \frac {A+B x}{x^{7/2} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {A}{4 b x^{7/2} \sqrt {b x+c x^2}}+\frac {\left (\frac {1}{2} (b B-2 A c)-\frac {7}{2} (-b B+A c)\right ) \int \frac {1}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx}{4 b}\\ &=-\frac {A}{4 b x^{7/2} \sqrt {b x+c x^2}}-\frac {8 b B-9 A c}{24 b^2 x^{5/2} \sqrt {b x+c x^2}}-\frac {(7 c (8 b B-9 A c)) \int \frac {1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx}{48 b^2}\\ &=-\frac {A}{4 b x^{7/2} \sqrt {b x+c x^2}}-\frac {8 b B-9 A c}{24 b^2 x^{5/2} \sqrt {b x+c x^2}}+\frac {7 c (8 b B-9 A c)}{96 b^3 x^{3/2} \sqrt {b x+c x^2}}+\frac {\left (35 c^2 (8 b B-9 A c)\right ) \int \frac {1}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx}{192 b^3}\\ &=-\frac {A}{4 b x^{7/2} \sqrt {b x+c x^2}}-\frac {8 b B-9 A c}{24 b^2 x^{5/2} \sqrt {b x+c x^2}}+\frac {7 c (8 b B-9 A c)}{96 b^3 x^{3/2} \sqrt {b x+c x^2}}-\frac {35 c^2 (8 b B-9 A c)}{192 b^4 \sqrt {x} \sqrt {b x+c x^2}}-\frac {\left (35 c^3 (8 b B-9 A c)\right ) \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{128 b^4}\\ &=-\frac {A}{4 b x^{7/2} \sqrt {b x+c x^2}}-\frac {8 b B-9 A c}{24 b^2 x^{5/2} \sqrt {b x+c x^2}}+\frac {7 c (8 b B-9 A c)}{96 b^3 x^{3/2} \sqrt {b x+c x^2}}-\frac {35 c^2 (8 b B-9 A c)}{192 b^4 \sqrt {x} \sqrt {b x+c x^2}}-\frac {35 c^3 (8 b B-9 A c) \sqrt {x}}{64 b^5 \sqrt {b x+c x^2}}-\frac {\left (35 c^3 (8 b B-9 A c)\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{128 b^5}\\ &=-\frac {A}{4 b x^{7/2} \sqrt {b x+c x^2}}-\frac {8 b B-9 A c}{24 b^2 x^{5/2} \sqrt {b x+c x^2}}+\frac {7 c (8 b B-9 A c)}{96 b^3 x^{3/2} \sqrt {b x+c x^2}}-\frac {35 c^2 (8 b B-9 A c)}{192 b^4 \sqrt {x} \sqrt {b x+c x^2}}-\frac {35 c^3 (8 b B-9 A c) \sqrt {x}}{64 b^5 \sqrt {b x+c x^2}}-\frac {\left (35 c^3 (8 b B-9 A c)\right ) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{64 b^5}\\ &=-\frac {A}{4 b x^{7/2} \sqrt {b x+c x^2}}-\frac {8 b B-9 A c}{24 b^2 x^{5/2} \sqrt {b x+c x^2}}+\frac {7 c (8 b B-9 A c)}{96 b^3 x^{3/2} \sqrt {b x+c x^2}}-\frac {35 c^2 (8 b B-9 A c)}{192 b^4 \sqrt {x} \sqrt {b x+c x^2}}-\frac {35 c^3 (8 b B-9 A c) \sqrt {x}}{64 b^5 \sqrt {b x+c x^2}}+\frac {35 c^3 (8 b B-9 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{64 b^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 153, normalized size = 0.71 \begin {gather*} \frac {\sqrt {b} \left (-8 b B x \left (8 b^3-14 b^2 c x+35 b c^2 x^2+105 c^3 x^3\right )+A \left (-48 b^4+72 b^3 c x-126 b^2 c^2 x^2+315 b c^3 x^3+945 c^4 x^4\right )\right )+105 c^3 (8 b B-9 A c) x^4 \sqrt {b+c x} \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{192 b^{11/2} x^{7/2} \sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(7/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

(Sqrt[b]*(-8*b*B*x*(8*b^3 - 14*b^2*c*x + 35*b*c^2*x^2 + 105*c^3*x^3) + A*(-48*b^4 + 72*b^3*c*x - 126*b^2*c^2*x
^2 + 315*b*c^3*x^3 + 945*c^4*x^4)) + 105*c^3*(8*b*B - 9*A*c)*x^4*Sqrt[b + c*x]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]])
/(192*b^(11/2)*x^(7/2)*Sqrt[x*(b + c*x)])

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Maple [A]
time = 0.57, size = 174, normalized size = 0.81

method	result	size
risch	\(-\frac {\left (c x +b \right ) \left (-561 A \,c^{3} x^{3}+456 B b \,c^{2} x^{3}+246 A b \,c^{2} x^{2}-176 B \,b^{2} c \,x^{2}-120 A \,b^{2} c x +64 B \,b^{3} x +48 A \,b^{3}\right )}{192 b^{5} x^{\frac {7}{2}} \sqrt {x \left (c x +b \right )}}+\frac {c^{3} \left (-\frac {2 \left (315 A c -280 B b \right ) \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \left (-128 A c +128 B b \right )}{\sqrt {c x +b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{128 b^{5} \sqrt {x \left (c x +b \right )}}\)	\(157\)
default	\(-\frac {\sqrt {x \left (c x +b \right )}\, \left (945 A \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, c^{4} x^{4}+64 B \,b^{\frac {9}{2}} x -112 B \,b^{\frac {7}{2}} c \,x^{2}+280 B \,b^{\frac {5}{2}} c^{2} x^{3}+840 B \,b^{\frac {3}{2}} c^{3} x^{4}-840 B \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, b \,c^{3} x^{4}+48 A \,b^{\frac {9}{2}}-72 A \,b^{\frac {7}{2}} c x +126 A \,b^{\frac {5}{2}} c^{2} x^{2}-315 A \,b^{\frac {3}{2}} c^{3} x^{3}-945 A \sqrt {b}\, c^{4} x^{4}\right )}{192 x^{\frac {9}{2}} \left (c x +b \right ) b^{\frac {11}{2}}}\)	\(174\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(7/2)/(c*x^2+b*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/192/x^(9/2)*(x*(c*x+b))^(1/2)*(945*A*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*c^4*x^4+64*B*b^(9/2)*x-11
2*B*b^(7/2)*c*x^2+280*B*b^(5/2)*c^2*x^3+840*B*b^(3/2)*c^3*x^4-840*B*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/
2)*b*c^3*x^4+48*A*b^(9/2)-72*A*b^(7/2)*c*x+126*A*b^(5/2)*c^2*x^2-315*A*b^(3/2)*c^3*x^3-945*A*b^(1/2)*c^4*x^4)/
(c*x+b)/b^(11/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(7/2)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(3/2)*x^(7/2)), x)

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Fricas [A]
time = 3.10, size = 406, normalized size = 1.88 \begin {gather*} \left [-\frac {105 \, {\left ({\left (8 \, B b c^{4} - 9 \, A c^{5}\right )} x^{6} + {\left (8 \, B b^{2} c^{3} - 9 \, A b c^{4}\right )} x^{5}\right )} \sqrt {b} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (48 \, A b^{5} + 105 \, {\left (8 \, B b^{2} c^{3} - 9 \, A b c^{4}\right )} x^{4} + 35 \, {\left (8 \, B b^{3} c^{2} - 9 \, A b^{2} c^{3}\right )} x^{3} - 14 \, {\left (8 \, B b^{4} c - 9 \, A b^{3} c^{2}\right )} x^{2} + 8 \, {\left (8 \, B b^{5} - 9 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{384 \, {\left (b^{6} c x^{6} + b^{7} x^{5}\right )}}, -\frac {105 \, {\left ({\left (8 \, B b c^{4} - 9 \, A c^{5}\right )} x^{6} + {\left (8 \, B b^{2} c^{3} - 9 \, A b c^{4}\right )} x^{5}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (48 \, A b^{5} + 105 \, {\left (8 \, B b^{2} c^{3} - 9 \, A b c^{4}\right )} x^{4} + 35 \, {\left (8 \, B b^{3} c^{2} - 9 \, A b^{2} c^{3}\right )} x^{3} - 14 \, {\left (8 \, B b^{4} c - 9 \, A b^{3} c^{2}\right )} x^{2} + 8 \, {\left (8 \, B b^{5} - 9 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{192 \, {\left (b^{6} c x^{6} + b^{7} x^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(7/2)/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

[-1/384*(105*((8*B*b*c^4 - 9*A*c^5)*x^6 + (8*B*b^2*c^3 - 9*A*b*c^4)*x^5)*sqrt(b)*log(-(c*x^2 + 2*b*x - 2*sqrt(
c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) + 2*(48*A*b^5 + 105*(8*B*b^2*c^3 - 9*A*b*c^4)*x^4 + 35*(8*B*b^3*c^2 - 9*A*b
^2*c^3)*x^3 - 14*(8*B*b^4*c - 9*A*b^3*c^2)*x^2 + 8*(8*B*b^5 - 9*A*b^4*c)*x)*sqrt(c*x^2 + b*x)*sqrt(x))/(b^6*c*
x^6 + b^7*x^5), -1/192*(105*((8*B*b*c^4 - 9*A*c^5)*x^6 + (8*B*b^2*c^3 - 9*A*b*c^4)*x^5)*sqrt(-b)*arctan(sqrt(-
b)*sqrt(x)/sqrt(c*x^2 + b*x)) + (48*A*b^5 + 105*(8*B*b^2*c^3 - 9*A*b*c^4)*x^4 + 35*(8*B*b^3*c^2 - 9*A*b^2*c^3)
*x^3 - 14*(8*B*b^4*c - 9*A*b^3*c^2)*x^2 + 8*(8*B*b^5 - 9*A*b^4*c)*x)*sqrt(c*x^2 + b*x)*sqrt(x))/(b^6*c*x^6 + b
^7*x^5)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{x^{\frac {7}{2}} \left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**(7/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral((A + B*x)/(x**(7/2)*(x*(b + c*x))**(3/2)), x)

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Giac [A]
time = 1.40, size = 197, normalized size = 0.91 \begin {gather*} -\frac {35 \, {\left (8 \, B b c^{3} - 9 \, A c^{4}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{64 \, \sqrt {-b} b^{5}} - \frac {2 \, {\left (B b c^{3} - A c^{4}\right )}}{\sqrt {c x + b} b^{5}} - \frac {456 \, {\left (c x + b\right )}^{\frac {7}{2}} B b c^{3} - 1544 \, {\left (c x + b\right )}^{\frac {5}{2}} B b^{2} c^{3} + 1784 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{3} c^{3} - 696 \, \sqrt {c x + b} B b^{4} c^{3} - 561 \, {\left (c x + b\right )}^{\frac {7}{2}} A c^{4} + 1929 \, {\left (c x + b\right )}^{\frac {5}{2}} A b c^{4} - 2295 \, {\left (c x + b\right )}^{\frac {3}{2}} A b^{2} c^{4} + 975 \, \sqrt {c x + b} A b^{3} c^{4}}{192 \, b^{5} c^{4} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(7/2)/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

-35/64*(8*B*b*c^3 - 9*A*c^4)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^5) - 2*(B*b*c^3 - A*c^4)/(sqrt(c*x + b
)*b^5) - 1/192*(456*(c*x + b)^(7/2)*B*b*c^3 - 1544*(c*x + b)^(5/2)*B*b^2*c^3 + 1784*(c*x + b)^(3/2)*B*b^3*c^3
- 696*sqrt(c*x + b)*B*b^4*c^3 - 561*(c*x + b)^(7/2)*A*c^4 + 1929*(c*x + b)^(5/2)*A*b*c^4 - 2295*(c*x + b)^(3/2
)*A*b^2*c^4 + 975*sqrt(c*x + b)*A*b^3*c^4)/(b^5*c^4*x^4)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^{7/2}\,{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(7/2)*(b*x + c*x^2)^(3/2)),x)

[Out]

int((A + B*x)/(x^(7/2)*(b*x + c*x^2)^(3/2)), x)

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