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

Optimal. Leaf size=128 \[ \frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \]

[Out]

(-2*(a + b*x)^(5/2))/(3*d*(c + d*x)^(3/2)) - (10*b*(a + b*x)^(3/2))/(3*d^2*Sqrt[c + d*x]) + (5*b^2*Sqrt[a + b*
x]*Sqrt[c + d*x])/d^3 - (5*b^(3/2)*(b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/d^(7/
2)

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Rubi [A]  time = 0.0667664, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {47, 50, 63, 217, 206} \[ \frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*x)^(5/2))/(3*d*(c + d*x)^(3/2)) - (10*b*(a + b*x)^(3/2))/(3*d^2*Sqrt[c + d*x]) + (5*b^2*Sqrt[a + b*
x]*Sqrt[c + d*x])/d^3 - (5*b^(3/2)*(b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/d^(7/
2)

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{(a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}+\frac{(5 b) \int \frac{(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx}{3 d}\\ &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}+\frac{\left (5 b^2\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{d^2}\\ &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{\left (5 b^2 (b c-a d)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 d^3}\\ &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{(5 b (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{d^3}\\ &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{(5 b (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{d^3}\\ &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0704367, size = 73, normalized size = 0.57 \[ \frac{2 (a+b x)^{7/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} \, _2F_1\left (\frac{5}{2},\frac{7}{2};\frac{9}{2};\frac{d (a+b x)}{a d-b c}\right )}{7 b (c+d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^(7/2)*((b*(c + d*x))/(b*c - a*d))^(5/2)*Hypergeometric2F1[5/2, 7/2, 9/2, (d*(a + b*x))/(-(b*c) +
a*d)])/(7*b*(c + d*x)^(5/2))

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Maple [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.77857, size = 1033, normalized size = 8.07 \begin{align*} \left [-\frac{15 \,{\left (b^{2} c^{3} - a b c^{2} d +{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} x\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \,{\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{12 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}, \frac{15 \,{\left (b^{2} c^{3} - a b c^{2} d +{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{-\frac{b}{d}}}{2 \,{\left (b^{2} d x^{2} + a b c +{\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \,{\left (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \,{\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(15*(b^2*c^3 - a*b*c^2*d + (b^2*c*d^2 - a*b*d^3)*x^2 + 2*(b^2*c^2*d - a*b*c*d^2)*x)*sqrt(b/d)*log(8*b^2
*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d)
+ 8*(b^2*c*d + a*b*d^2)*x) - 4*(3*b^2*d^2*x^2 + 15*b^2*c^2 - 10*a*b*c*d - 2*a^2*d^2 + 2*(10*b^2*c*d - 7*a*b*d
^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3), 1/6*(15*(b^2*c^3 - a*b*c^2*d + (b^2*c*d^2
- a*b*d^3)*x^2 + 2*(b^2*c^2*d - a*b*c*d^2)*x)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(
d*x + c)*sqrt(-b/d)/(b^2*d*x^2 + a*b*c + (b^2*c + a*b*d)*x)) + 2*(3*b^2*d^2*x^2 + 15*b^2*c^2 - 10*a*b*c*d - 2*
a^2*d^2 + 2*(10*b^2*c*d - 7*a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)]

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

[Out]

Timed out

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Giac [B]  time = 2.32651, size = 373, normalized size = 2.91 \begin{align*} \frac{{\left ({\left (b x + a\right )}{\left (\frac{3 \,{\left (b^{6} c d^{4} - a b^{5} d^{5}\right )}{\left (b x + a\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}} + \frac{20 \,{\left (b^{7} c^{2} d^{3} - 2 \, a b^{6} c d^{4} + a^{2} b^{5} d^{5}\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}}\right )} + \frac{15 \,{\left (b^{8} c^{3} d^{2} - 3 \, a b^{7} c^{2} d^{3} + 3 \, a^{2} b^{6} c d^{4} - a^{3} b^{5} d^{5}\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}}\right )} \sqrt{b x + a}}{3 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{5 \,{\left (b^{4} c - a b^{3} d\right )} \log \left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{3}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((b*x + a)*(3*(b^6*c*d^4 - a*b^5*d^5)*(b*x + a)/(b^2*c*d^5*abs(b) - a*b*d^6*abs(b)) + 20*(b^7*c^2*d^3 - 2*
a*b^6*c*d^4 + a^2*b^5*d^5)/(b^2*c*d^5*abs(b) - a*b*d^6*abs(b))) + 15*(b^8*c^3*d^2 - 3*a*b^7*c^2*d^3 + 3*a^2*b^
6*c*d^4 - a^3*b^5*d^5)/(b^2*c*d^5*abs(b) - a*b*d^6*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2
) + 5*(b^4*c - a*b^3*d)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d^
3*abs(b))