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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.194758, antiderivative size = 333, normalized size of antiderivative = 1., 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 = {96, 94, 93, 208} \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+7 b c) (b c-a d)^3}{768 a^3 c^3 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (5 a d+7 b c) (b c-a d)^2}{960 a^2 c^3 x^3}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^4}{512 a^4 c^3 x}-\frac{(5 a d+7 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{9/2} c^{7/2}}+\frac{\sqrt{a+b x} (c+d x)^{7/2} (5 a d+7 b c) (b c-a d)}{160 a c^3 x^4}+\frac{(a+b x)^{3/2} (c+d x)^{7/2} (5 a d+7 b c)}{60 a c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6} \]

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 96

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 94

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[p, 1] &&  !SumSimplerQ[m, 1])

Rule 93

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 208

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

Mathematica [A]  time = 0.765828, size = 272, normalized size = 0.82 \[ \frac{(5 a d+7 b c) \left (128 a^{7/2} c^{3/2} (a+b x)^{3/2} (c+d x)^{7/2}+x (b c-a d) \left (x (b c-a d) \left (8 a^{5/2} \sqrt{c} \sqrt{a+b x} (c+d x)^{5/2}-5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )\right )+48 a^{7/2} \sqrt{c} \sqrt{a+b x} (c+d x)^{7/2}\right )\right )}{7680 a^{9/2} c^{7/2} x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x)^(5/2)*(c + d*x)^(7/2))/(6*a*c*x^6) + ((7*b*c + 5*a*d)*(128*a^(7/2)*c^(3/2)*(a + b*x)^(3/2)*(c + d*
x)^(7/2) + (b*c - a*d)*x*(48*a^(7/2)*Sqrt[c]*Sqrt[a + b*x]*(c + d*x)^(7/2) + (b*c - a*d)*x*(8*a^(5/2)*Sqrt[c]*
Sqrt[a + b*x]*(c + d*x)^(5/2) - 5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c - 3*b*c*x
+ 5*a*d*x) + 3*(b*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])))))/(7680*a^(9/2)*c
^(7/2)*x^5)

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Maple [B]  time = 0.021, size = 1271, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^3*(210*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*b^5*c^5+100*(
a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^5*c*d^4-140*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a
*b^4*c^5-80*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^5*c^2*d^3-3328*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*x*a^4*b*c^5-270*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^5*b*c
*d^5+225*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^4*b^2*c^2*d^4+300*ln((a
*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^3*b^3*c^3*d^3-675*ln((a*d*x+b*c*x+2*(
a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^2*b^4*c^4*d^2+450*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a*b^5*c^5*d+112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2
*b^3*c^5-96*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*b^2*c^5-4320*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*x^2*a^5*c^3*d^2-150*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^5*d^5-6400*(a*c)^(1/2)*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^5*c^4*d-2560*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*c^5*(a*c)^(1/2)+75*ln((a*d*x+b*
c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^6*d^6-105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d
*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*b^6*c^6+490*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^4*b*c*
d^4-300*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^3*b^2*c^2*d^3+1092*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*x^5*a^2*b^3*c^3*d^2-830*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a*b^4*c^4*d-320*(a*c)^(1/2)
*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^4*b*c^2*d^3-696*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^3*b^2
*c^3*d^2+544*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^2*b^3*c^4*d-6768*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*x^3*a^4*b*c^3*d^2-432*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*b^2*c^4*d-8896*(a*c)^(
1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^4*b*c^4*d)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^6/(a*c)^(1/2)

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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)^(3/2)*(d*x+c)^(5/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

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]

Exception raised: UnboundLocalError

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

[Out]

Timed out

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

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]

Exception raised: TypeError