3.7.25 \(\int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx\) [625]

Optimal. Leaf size=333 \[ \frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 a^4 c^3 x}-\frac {(b c-a d)^3 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{768 a^3 c^3 x^2}+\frac {(b c-a d)^2 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{960 a^2 c^3 x^3}+\frac {(b c-a d) (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{7/2}}{160 a c^3 x^4}+\frac {(7 b c+5 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 a c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}-\frac {(b c-a d)^5 (7 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{9/2} c^{7/2}} \]

[Out]

1/60*(5*a*d+7*b*c)*(b*x+a)^(3/2)*(d*x+c)^(7/2)/a/c^2/x^5-1/6*(b*x+a)^(5/2)*(d*x+c)^(7/2)/a/c/x^6-1/512*(-a*d+b
*c)^5*(5*a*d+7*b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(9/2)/c^(7/2)-1/768*(-a*d+b*c)^3*(5
*a*d+7*b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a^3/c^3/x^2+1/960*(-a*d+b*c)^2*(5*a*d+7*b*c)*(d*x+c)^(5/2)*(b*x+a)^(1/
2)/a^2/c^3/x^3+1/160*(-a*d+b*c)*(5*a*d+7*b*c)*(d*x+c)^(7/2)*(b*x+a)^(1/2)/a/c^3/x^4+1/512*(-a*d+b*c)^4*(5*a*d+
7*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^3/x

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Rubi [A]
time = 0.14, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {98, 96, 95, 214} \begin {gather*} -\frac {(5 a d+7 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{9/2} c^{7/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)^4}{512 a^4 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+7 b c) (b c-a d)^3}{768 a^3 c^3 x^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+7 b c) (b c-a d)^2}{960 a^2 c^3 x^3}+\frac {\sqrt {a+b x} (c+d x)^{7/2} (5 a d+7 b c) (b c-a d)}{160 a c^3 x^4}+\frac {(a+b x)^{3/2} (c+d x)^{7/2} (5 a d+7 b c)}{60 a c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^7,x]

[Out]

((b*c - a*d)^4*(7*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*a^4*c^3*x) - ((b*c - a*d)^3*(7*b*c + 5*a*d)*S
qrt[a + b*x]*(c + d*x)^(3/2))/(768*a^3*c^3*x^2) + ((b*c - a*d)^2*(7*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2)
)/(960*a^2*c^3*x^3) + ((b*c - a*d)*(7*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/(160*a*c^3*x^4) + ((7*b*c +
5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(60*a*c^2*x^5) - ((a + b*x)^(5/2)*(c + d*x)^(7/2))/(6*a*c*x^6) - ((b*c
- a*d)^5*(7*b*c + 5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(512*a^(9/2)*c^(7/2))

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

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

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 214

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

Rubi steps

\begin {align*} \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}-\frac {\left (\frac {7 b c}{2}+\frac {5 a d}{2}\right ) \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx}{6 a c}\\ &=\frac {(7 b c+5 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 a c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}-\frac {((b c-a d) (7 b c+5 a d)) \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^5} \, dx}{40 a c^2}\\ &=\frac {(b c-a d) (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{7/2}}{160 a c^3 x^4}+\frac {(7 b c+5 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 a c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\right ) \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}} \, dx}{320 a c^3}\\ &=\frac {(b c-a d)^2 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{960 a^2 c^3 x^3}+\frac {(b c-a d) (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{7/2}}{160 a c^3 x^4}+\frac {(7 b c+5 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 a c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}+\frac {\left ((b c-a d)^3 (7 b c+5 a d)\right ) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx}{384 a^2 c^3}\\ &=-\frac {(b c-a d)^3 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{768 a^3 c^3 x^2}+\frac {(b c-a d)^2 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{960 a^2 c^3 x^3}+\frac {(b c-a d) (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{7/2}}{160 a c^3 x^4}+\frac {(7 b c+5 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 a c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}-\frac {\left ((b c-a d)^4 (7 b c+5 a d)\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{512 a^3 c^3}\\ &=\frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 a^4 c^3 x}-\frac {(b c-a d)^3 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{768 a^3 c^3 x^2}+\frac {(b c-a d)^2 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{960 a^2 c^3 x^3}+\frac {(b c-a d) (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{7/2}}{160 a c^3 x^4}+\frac {(7 b c+5 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 a c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}+\frac {\left ((b c-a d)^5 (7 b c+5 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{1024 a^4 c^3}\\ &=\frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 a^4 c^3 x}-\frac {(b c-a d)^3 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{768 a^3 c^3 x^2}+\frac {(b c-a d)^2 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{960 a^2 c^3 x^3}+\frac {(b c-a d) (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{7/2}}{160 a c^3 x^4}+\frac {(7 b c+5 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 a c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}+\frac {\left ((b c-a d)^5 (7 b c+5 a d)\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{512 a^4 c^3}\\ &=\frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 a^4 c^3 x}-\frac {(b c-a d)^3 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{768 a^3 c^3 x^2}+\frac {(b c-a d)^2 (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{960 a^2 c^3 x^3}+\frac {(b c-a d) (7 b c+5 a d) \sqrt {a+b x} (c+d x)^{7/2}}{160 a c^3 x^4}+\frac {(7 b c+5 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 a c^2 x^5}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6}-\frac {(b c-a d)^5 (7 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{9/2} c^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.72, size = 322, normalized size = 0.97 \begin {gather*} \frac {(-b c+a d)^5 \left (\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} \left (-105 b^5 c^5 x^5+5 a b^4 c^4 x^4 (14 c+83 d x)-2 a^2 b^3 c^3 x^3 \left (28 c^2+136 c d x+273 d^2 x^2\right )+6 a^3 b^2 c^2 x^2 \left (8 c^3+36 c^2 d x+58 c d^2 x^2+25 d^3 x^3\right )+a^4 b c x \left (1664 c^4+4448 c^3 d x+3384 c^2 d^2 x^2+160 c d^3 x^3-245 d^4 x^4\right )+5 a^5 \left (256 c^5+640 c^4 d x+432 c^3 d^2 x^2+8 c^2 d^3 x^3-10 c d^4 x^4+15 d^5 x^5\right )\right )}{(b c-a d)^5 x^6}+15 (7 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )}{7680 a^{9/2} c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^7,x]

[Out]

((-(b*c) + a*d)^5*((Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*b^5*c^5*x^5 + 5*a*b^4*c^4*x^4*(14*c + 83
*d*x) - 2*a^2*b^3*c^3*x^3*(28*c^2 + 136*c*d*x + 273*d^2*x^2) + 6*a^3*b^2*c^2*x^2*(8*c^3 + 36*c^2*d*x + 58*c*d^
2*x^2 + 25*d^3*x^3) + a^4*b*c*x*(1664*c^4 + 4448*c^3*d*x + 3384*c^2*d^2*x^2 + 160*c*d^3*x^3 - 245*d^4*x^4) + 5
*a^5*(256*c^5 + 640*c^4*d*x + 432*c^3*d^2*x^2 + 8*c^2*d^3*x^3 - 10*c*d^4*x^4 + 15*d^5*x^5)))/((b*c - a*d)^5*x^
6) + 15*(7*b*c + 5*a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])]))/(7680*a^(9/2)*c^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1067\) vs. \(2(283)=566\).
time = 0.07, size = 1068, normalized size = 3.21

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (490 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b c \,d^{4} x^{5}-300 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c^{2} d^{3} x^{5}+1092 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{3} d^{2} x^{5}-830 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{4} c^{4} d \,x^{5}-270 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{5} b c \,d^{5} x^{6}+225 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} b^{2} c^{2} d^{4} x^{6}+300 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b^{3} c^{3} d^{3} x^{6}-675 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{4} c^{4} d^{2} x^{6}+450 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{5} c^{5} d \,x^{6}-80 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} c^{2} d^{3} x^{3}+112 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{5} x^{3}-4320 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} c^{3} d^{2} x^{2}-6400 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} c^{4} d x -3328 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b \,c^{5} x +100 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} c \,d^{4} x^{4}-140 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{4} c^{5} x^{4}-96 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c^{5} x^{2}-2560 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} c^{5} \sqrt {a c}+75 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{6} d^{6} x^{6}-105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{6} c^{6} x^{6}-320 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b \,c^{2} d^{3} x^{4}-696 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c^{3} d^{2} x^{4}+544 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{3} c^{4} d \,x^{4}-6768 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b \,c^{3} d^{2} x^{3}-432 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b^{2} c^{4} d \,x^{3}-8896 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} b \,c^{4} d \,x^{2}+210 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{5} c^{5} x^{5}-150 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{5} d^{5} x^{5}\right )}{15360 a^{4} c^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{6} \sqrt {a c}}\)	\(1068\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^3*(490*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b*c*d^4*x^5-300*(a*c)
^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b^2*c^2*d^3*x^5+1092*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^3*c^3*d^2*x^
5-830*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^4*c^4*d*x^5-270*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^
(1/2)+2*a*c)/x)*a^5*b*c*d^5*x^6+225*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^4*b^2*c^
2*d^4*x^6+300*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*b^3*c^3*d^3*x^6-675*ln((a*d*
x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b^4*c^4*d^2*x^6+450*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*
((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a*b^5*c^5*d*x^6-80*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^5*c^2*d^3*x^3+112*(
a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^3*c^5*x^3-4320*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^5*c^3*d^2*x^2-64
00*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^5*c^4*d*x-3328*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b*c^5*x+100*(a
*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^5*c*d^4*x^4-140*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^4*c^5*x^4-96*(a*c)
^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b^2*c^5*x^2-2560*((d*x+c)*(b*x+a))^(1/2)*a^5*c^5*(a*c)^(1/2)+75*ln((a*d*x+b
*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^6*d^6*x^6-105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b
*x+a))^(1/2)+2*a*c)/x)*b^6*c^6*x^6-320*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b*c^2*d^3*x^4-696*(a*c)^(1/2)*(
(d*x+c)*(b*x+a))^(1/2)*a^3*b^2*c^3*d^2*x^4+544*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^3*c^4*d*x^4-6768*(a*c
)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b*c^3*d^2*x^3-432*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b^2*c^4*d*x^3-88
96*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b*c^4*d*x^2+210*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^5*c^5*x^5-150
*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^5*d^5*x^5)/((d*x+c)*(b*x+a))^(1/2)/x^6/(a*c)^(1/2)

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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)^(3/2)*(d*x+c)^(5/2)/x^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*d-b*c>0)', see `assume?` for
more detail

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Fricas [A]
time = 10.20, size = 922, normalized size = 2.77 \begin {gather*} \left [-\frac {15 \, {\left (7 \, b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 45 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 15 \, a^{4} b^{2} c^{2} d^{4} + 18 \, a^{5} b c d^{5} - 5 \, a^{6} d^{6}\right )} \sqrt {a c} x^{6} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (1280 \, a^{6} c^{6} - {\left (105 \, a b^{5} c^{6} - 415 \, a^{2} b^{4} c^{5} d + 546 \, a^{3} b^{3} c^{4} d^{2} - 150 \, a^{4} b^{2} c^{3} d^{3} + 245 \, a^{5} b c^{2} d^{4} - 75 \, a^{6} c d^{5}\right )} x^{5} + 2 \, {\left (35 \, a^{2} b^{4} c^{6} - 136 \, a^{3} b^{3} c^{5} d + 174 \, a^{4} b^{2} c^{4} d^{2} + 80 \, a^{5} b c^{3} d^{3} - 25 \, a^{6} c^{2} d^{4}\right )} x^{4} - 8 \, {\left (7 \, a^{3} b^{3} c^{6} - 27 \, a^{4} b^{2} c^{5} d - 423 \, a^{5} b c^{4} d^{2} - 5 \, a^{6} c^{3} d^{3}\right )} x^{3} + 16 \, {\left (3 \, a^{4} b^{2} c^{6} + 278 \, a^{5} b c^{5} d + 135 \, a^{6} c^{4} d^{2}\right )} x^{2} + 128 \, {\left (13 \, a^{5} b c^{6} + 25 \, a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, a^{5} c^{4} x^{6}}, \frac {15 \, {\left (7 \, b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 45 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 15 \, a^{4} b^{2} c^{2} d^{4} + 18 \, a^{5} b c d^{5} - 5 \, a^{6} d^{6}\right )} \sqrt {-a c} x^{6} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (1280 \, a^{6} c^{6} - {\left (105 \, a b^{5} c^{6} - 415 \, a^{2} b^{4} c^{5} d + 546 \, a^{3} b^{3} c^{4} d^{2} - 150 \, a^{4} b^{2} c^{3} d^{3} + 245 \, a^{5} b c^{2} d^{4} - 75 \, a^{6} c d^{5}\right )} x^{5} + 2 \, {\left (35 \, a^{2} b^{4} c^{6} - 136 \, a^{3} b^{3} c^{5} d + 174 \, a^{4} b^{2} c^{4} d^{2} + 80 \, a^{5} b c^{3} d^{3} - 25 \, a^{6} c^{2} d^{4}\right )} x^{4} - 8 \, {\left (7 \, a^{3} b^{3} c^{6} - 27 \, a^{4} b^{2} c^{5} d - 423 \, a^{5} b c^{4} d^{2} - 5 \, a^{6} c^{3} d^{3}\right )} x^{3} + 16 \, {\left (3 \, a^{4} b^{2} c^{6} + 278 \, a^{5} b c^{5} d + 135 \, a^{6} c^{4} d^{2}\right )} x^{2} + 128 \, {\left (13 \, a^{5} b c^{6} + 25 \, a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, a^{5} c^{4} x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^7,x, algorithm="fricas")

[Out]

[-1/30720*(15*(7*b^6*c^6 - 30*a*b^5*c^5*d + 45*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 15*a^4*b^2*c^2*d^4 + 18*
a^5*b*c*d^5 - 5*a^6*d^6)*sqrt(a*c)*x^6*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c + (b*c
+ a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(1280*a^6*c^6 - (105*a*b^5
*c^6 - 415*a^2*b^4*c^5*d + 546*a^3*b^3*c^4*d^2 - 150*a^4*b^2*c^3*d^3 + 245*a^5*b*c^2*d^4 - 75*a^6*c*d^5)*x^5 +
2*(35*a^2*b^4*c^6 - 136*a^3*b^3*c^5*d + 174*a^4*b^2*c^4*d^2 + 80*a^5*b*c^3*d^3 - 25*a^6*c^2*d^4)*x^4 - 8*(7*a
^3*b^3*c^6 - 27*a^4*b^2*c^5*d - 423*a^5*b*c^4*d^2 - 5*a^6*c^3*d^3)*x^3 + 16*(3*a^4*b^2*c^6 + 278*a^5*b*c^5*d +
135*a^6*c^4*d^2)*x^2 + 128*(13*a^5*b*c^6 + 25*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^4*x^6), 1/153
60*(15*(7*b^6*c^6 - 30*a*b^5*c^5*d + 45*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 15*a^4*b^2*c^2*d^4 + 18*a^5*b*c
*d^5 - 5*a^6*d^6)*sqrt(-a*c)*x^6*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*
b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) - 2*(1280*a^6*c^6 - (105*a*b^5*c^6 - 415*a^2*b^4*c^5*d + 546*a^3
*b^3*c^4*d^2 - 150*a^4*b^2*c^3*d^3 + 245*a^5*b*c^2*d^4 - 75*a^6*c*d^5)*x^5 + 2*(35*a^2*b^4*c^6 - 136*a^3*b^3*c
^5*d + 174*a^4*b^2*c^4*d^2 + 80*a^5*b*c^3*d^3 - 25*a^6*c^2*d^4)*x^4 - 8*(7*a^3*b^3*c^6 - 27*a^4*b^2*c^5*d - 42
3*a^5*b*c^4*d^2 - 5*a^6*c^3*d^3)*x^3 + 16*(3*a^4*b^2*c^6 + 278*a^5*b*c^5*d + 135*a^6*c^4*d^2)*x^2 + 128*(13*a^
5*b*c^6 + 25*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^4*x^6)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(d*x+c)**(5/2)/x**7,x)

[Out]

Integral((a + b*x)**(3/2)*(c + d*x)**(5/2)/x**7, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 8502 vs. \(2 (283) = 566\).
time = 7.05, size = 8502, normalized size = 25.53 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^7,x, algorithm="giac")

[Out]

-1/7680*(15*(7*sqrt(b*d)*b^7*c^6*abs(b) - 30*sqrt(b*d)*a*b^6*c^5*d*abs(b) + 45*sqrt(b*d)*a^2*b^5*c^4*d^2*abs(b
) - 20*sqrt(b*d)*a^3*b^4*c^3*d^3*abs(b) - 15*sqrt(b*d)*a^4*b^3*c^2*d^4*abs(b) + 18*sqrt(b*d)*a^5*b^2*c*d^5*abs
(b) - 5*sqrt(b*d)*a^6*b*d^6*abs(b))*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^4*b*c^3) - 2*(105*sqrt(b*d)*b^29*c^17*abs(b) - 167
5*sqrt(b*d)*a*b^28*c^16*d*abs(b) + 12456*sqrt(b*d)*a^2*b^27*c^15*d^2*abs(b) - 57192*sqrt(b*d)*a^3*b^26*c^14*d^
3*abs(b) + 181356*sqrt(b*d)*a^4*b^25*c^13*d^4*abs(b) - 421620*sqrt(b*d)*a^5*b^24*c^12*d^5*abs(b) + 746040*sqrt
(b*d)*a^6*b^23*c^11*d^6*abs(b) - 1032152*sqrt(b*d)*a^7*b^22*c^10*d^7*abs(b) + 1141734*sqrt(b*d)*a^8*b^21*c^9*d
^8*abs(b) - 1030722*sqrt(b*d)*a^9*b^20*c^8*d^9*abs(b) + 773080*sqrt(b*d)*a^10*b^19*c^7*d^10*abs(b) - 486360*sq
rt(b*d)*a^11*b^18*c^6*d^11*abs(b) + 254796*sqrt(b*d)*a^12*b^17*c^5*d^12*abs(b) - 107892*sqrt(b*d)*a^13*b^16*c^
4*d^13*abs(b) + 35016*sqrt(b*d)*a^14*b^15*c^3*d^14*abs(b) - 8040*sqrt(b*d)*a^15*b^14*c^2*d^15*abs(b) + 1145*sq
rt(b*d)*a^16*b^13*c*d^16*abs(b) - 75*sqrt(b*d)*a^17*b^12*d^17*abs(b) - 1155*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^27*c^16*abs(b) + 14820*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^26*c^15*d*abs(b) - 85572*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2*a^2*b^25*c^14*d^2*abs(b) + 289668*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2*a^3*b^24*c^13*d^3*abs(b) - 626160*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2*a^4*b^23*c^12*d^4*abs(b) + 869940*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2*a^5*b^22*c^11*d^5*abs(b) - 691620*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2*a^6*b^21*c^10*d^6*abs(b) + 66324*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^7*b^20*c^9*d^7*abs(b) + 603234*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^8*b^19*c^8*d^8*abs(b) - 921300*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^2*a^9*b^18*c^7*d^9*abs(b) + 864660*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^2*a^10*b^17*c^6*d^10*abs(b) - 630900*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^2*a^11*b^16*c^5*d^11*abs(b) + 373848*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^12*b^15*c^4*d^12*abs(b) - 170052*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^13*b^14*c^3*d^13*abs(b) + 53460*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^14*b^13*c^2*d^14*abs(b) - 10020*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^15*b^12*c*d^15*abs(b) + 825*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^2*a^16*b^11*d^16*abs(b) + 5775*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a
)*b*d - a*b*d))^4*b^25*c^15*abs(b) - 58455*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*a*b^24*c^14*d*abs(b) + 255411*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^4*a^2*b^23*c^13*d^2*abs(b) - 614355*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^4*a^3*b^22*c^12*d^3*abs(b) + 839955*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^4*a^4*b^21*c^11*d^4*abs(b) - 524715*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^4*a^5*b^20*c^10*d^5*abs(b) - 211305*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^4*a^6*b^19*c^9*d^6*abs(b) + 722073*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^4*a^7*b^18*c^8*d^7*abs(b) - 649635*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^4*a^8*b^17*c^7*d^8*abs(b) + 225195*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
4*a^9*b^16*c^6*d^9*abs(b) + 188265*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4
*a^10*b^15*c^5*d^10*abs(b) - 380265*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
4*a^11*b^14*c^4*d^11*abs(b) + 319041*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^4*a^12*b^13*c^3*d^12*abs(b) - 151785*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^4*a^13*b^12*c^2*d^13*abs(b) + 38925*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^4*a^14*b^11*c*d^14*abs(b) - 4125*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4
*a^15*b^10*d^15*abs(b) - 17325*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^2
3*c^14*abs(b) + 135570*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^22*c^13
*d*abs(b) - 435915*sqrt(b*d)*(sqrt(b*d)*sqrt(b*...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^7,x)

[Out]

int(((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^7, x)

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