3.1512 \(\int \frac{(a+b x)^{9/2}}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=204 \[ \frac{7 b^2 (a+b x)^{5/2} \sqrt{c+d x}}{d^3}-\frac{35 b^2 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{4 d^4}+\frac{105 b^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 d^5}-\frac{105 b^{3/2} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 d^{11/2}}-\frac{6 b (a+b x)^{7/2}}{d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}} \]

[Out]

(-2*(a + b*x)^(9/2))/(3*d*(c + d*x)^(3/2)) - (6*b*(a + b*x)^(7/2))/(d^2*Sqrt[c + d*x]) + (105*b^2*(b*c - a*d)^
2*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*d^5) - (35*b^2*(b*c - a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(4*d^4) + (7*b^2*(
a + b*x)^(5/2)*Sqrt[c + d*x])/d^3 - (105*b^(3/2)*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c
+ d*x])])/(8*d^(11/2))

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Rubi [A]  time = 0.108661, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{7 b^2 (a+b x)^{5/2} \sqrt{c+d x}}{d^3}-\frac{35 b^2 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{4 d^4}+\frac{105 b^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 d^5}-\frac{105 b^{3/2} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 d^{11/2}}-\frac{6 b (a+b x)^{7/2}}{d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*x)^(9/2))/(3*d*(c + d*x)^(3/2)) - (6*b*(a + b*x)^(7/2))/(d^2*Sqrt[c + d*x]) + (105*b^2*(b*c - a*d)^
2*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*d^5) - (35*b^2*(b*c - a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(4*d^4) + (7*b^2*(
a + b*x)^(5/2)*Sqrt[c + d*x])/d^3 - (105*b^(3/2)*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c
+ d*x])])/(8*d^(11/2))

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)^{9/2}}{(c+d x)^{5/2}} \, dx &=-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}+\frac{(3 b) \int \frac{(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx}{d}\\ &=-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac{6 b (a+b x)^{7/2}}{d^2 \sqrt{c+d x}}+\frac{\left (21 b^2\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx}{d^2}\\ &=-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac{6 b (a+b x)^{7/2}}{d^2 \sqrt{c+d x}}+\frac{7 b^2 (a+b x)^{5/2} \sqrt{c+d x}}{d^3}-\frac{\left (35 b^2 (b c-a d)\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{2 d^3}\\ &=-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac{6 b (a+b x)^{7/2}}{d^2 \sqrt{c+d x}}-\frac{35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{4 d^4}+\frac{7 b^2 (a+b x)^{5/2} \sqrt{c+d x}}{d^3}+\frac{\left (105 b^2 (b c-a d)^2\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{8 d^4}\\ &=-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac{6 b (a+b x)^{7/2}}{d^2 \sqrt{c+d x}}+\frac{105 b^2 (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^5}-\frac{35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{4 d^4}+\frac{7 b^2 (a+b x)^{5/2} \sqrt{c+d x}}{d^3}-\frac{\left (105 b^2 (b c-a d)^3\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 d^5}\\ &=-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac{6 b (a+b x)^{7/2}}{d^2 \sqrt{c+d x}}+\frac{105 b^2 (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^5}-\frac{35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{4 d^4}+\frac{7 b^2 (a+b x)^{5/2} \sqrt{c+d x}}{d^3}-\frac{\left (105 b (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{8 d^5}\\ &=-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac{6 b (a+b x)^{7/2}}{d^2 \sqrt{c+d x}}+\frac{105 b^2 (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^5}-\frac{35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{4 d^4}+\frac{7 b^2 (a+b x)^{5/2} \sqrt{c+d x}}{d^3}-\frac{\left (105 b (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{8 d^5}\\ &=-\frac{2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac{6 b (a+b x)^{7/2}}{d^2 \sqrt{c+d x}}+\frac{105 b^2 (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^5}-\frac{35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{4 d^4}+\frac{7 b^2 (a+b x)^{5/2} \sqrt{c+d x}}{d^3}-\frac{105 b^{3/2} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 d^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.096833, size = 73, normalized size = 0.36 \[ \frac{2 (a+b x)^{11/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} \, _2F_1\left (\frac{5}{2},\frac{11}{2};\frac{13}{2};\frac{d (a+b x)}{a d-b c}\right )}{11 b (c+d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^(11/2)*((b*(c + d*x))/(b*c - a*d))^(5/2)*Hypergeometric2F1[5/2, 11/2, 13/2, (d*(a + b*x))/(-(b*c)
+ a*d)])/(11*b*(c + d*x)^(5/2))

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{9}{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)^(9/2)/(d*x+c)^(5/2),x)

[Out]

int((b*x+a)^(9/2)/(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)^(9/2)/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 16.1911, size = 1874, normalized size = 9.19 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

[-1/96*(315*(b^4*c^5 - 3*a*b^3*c^4*d + 3*a^2*b^2*c^3*d^2 - a^3*b*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*
a^2*b^2*c*d^4 - a^3*b*d^5)*x^2 + 2*(b^4*c^4*d - 3*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 - a^3*b*c*d^4)*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*(8*b^4*d^4*x^4 + 315*b^4*c^4 - 840*a*b^3*c^3*d + 693*a^2*b^2*c^2*d^
2 - 144*a^3*b*c*d^3 - 16*a^4*d^4 - 2*(9*b^4*c*d^3 - 25*a*b^3*d^4)*x^3 + 3*(21*b^4*c^2*d^2 - 60*a*b^3*c*d^3 + 5
5*a^2*b^2*d^4)*x^2 + 2*(210*b^4*c^3*d - 567*a*b^3*c^2*d^2 + 477*a^2*b^2*c*d^3 - 104*a^3*b*d^4)*x)*sqrt(b*x + a
)*sqrt(d*x + c))/(d^7*x^2 + 2*c*d^6*x + c^2*d^5), 1/48*(315*(b^4*c^5 - 3*a*b^3*c^4*d + 3*a^2*b^2*c^3*d^2 - a^3
*b*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^2 + 2*(b^4*c^4*d - 3*a*b^3*c^3*d^
2 + 3*a^2*b^2*c^2*d^3 - a^3*b*c*d^4)*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*(8*b^4*d^4*x^4 + 315*b^4*c^4 - 840*a*b^3*c^3*d + 693
*a^2*b^2*c^2*d^2 - 144*a^3*b*c*d^3 - 16*a^4*d^4 - 2*(9*b^4*c*d^3 - 25*a*b^3*d^4)*x^3 + 3*(21*b^4*c^2*d^2 - 60*
a*b^3*c*d^3 + 55*a^2*b^2*d^4)*x^2 + 2*(210*b^4*c^3*d - 567*a*b^3*c^2*d^2 + 477*a^2*b^2*c*d^3 - 104*a^3*b*d^4)*
x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^7*x^2 + 2*c*d^6*x + c^2*d^5)]

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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)**(9/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.25007, size = 675, normalized size = 3.31 \begin{align*} \frac{{\left ({\left ({\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b^{6} c d^{8} - a b^{5} d^{9}\right )}{\left (b x + a\right )}}{b^{2} c d^{9}{\left | b \right |} - a b d^{10}{\left | b \right |}} - \frac{9 \,{\left (b^{7} c^{2} d^{7} - 2 \, a b^{6} c d^{8} + a^{2} b^{5} d^{9}\right )}}{b^{2} c d^{9}{\left | b \right |} - a b d^{10}{\left | b \right |}}\right )} + \frac{63 \,{\left (b^{8} c^{3} d^{6} - 3 \, a b^{7} c^{2} d^{7} + 3 \, a^{2} b^{6} c d^{8} - a^{3} b^{5} d^{9}\right )}}{b^{2} c d^{9}{\left | b \right |} - a b d^{10}{\left | b \right |}}\right )}{\left (b x + a\right )} + \frac{420 \,{\left (b^{9} c^{4} d^{5} - 4 \, a b^{8} c^{3} d^{6} + 6 \, a^{2} b^{7} c^{2} d^{7} - 4 \, a^{3} b^{6} c d^{8} + a^{4} b^{5} d^{9}\right )}}{b^{2} c d^{9}{\left | b \right |} - a b d^{10}{\left | b \right |}}\right )}{\left (b x + a\right )} + \frac{315 \,{\left (b^{10} c^{5} d^{4} - 5 \, a b^{9} c^{4} d^{5} + 10 \, a^{2} b^{8} c^{3} d^{6} - 10 \, a^{3} b^{7} c^{2} d^{7} + 5 \, a^{4} b^{6} c d^{8} - a^{5} b^{5} d^{9}\right )}}{b^{2} c d^{9}{\left | b \right |} - a b d^{10}{\left | b \right |}}\right )} \sqrt{b x + a}}{24 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{105 \,{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\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 )}{8 \, \sqrt{b d} d^{5}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

1/24*(((2*(b*x + a)*(4*(b^6*c*d^8 - a*b^5*d^9)*(b*x + a)/(b^2*c*d^9*abs(b) - a*b*d^10*abs(b)) - 9*(b^7*c^2*d^7
- 2*a*b^6*c*d^8 + a^2*b^5*d^9)/(b^2*c*d^9*abs(b) - a*b*d^10*abs(b))) + 63*(b^8*c^3*d^6 - 3*a*b^7*c^2*d^7 + 3*
a^2*b^6*c*d^8 - a^3*b^5*d^9)/(b^2*c*d^9*abs(b) - a*b*d^10*abs(b)))*(b*x + a) + 420*(b^9*c^4*d^5 - 4*a*b^8*c^3*
d^6 + 6*a^2*b^7*c^2*d^7 - 4*a^3*b^6*c*d^8 + a^4*b^5*d^9)/(b^2*c*d^9*abs(b) - a*b*d^10*abs(b)))*(b*x + a) + 315
*(b^10*c^5*d^4 - 5*a*b^9*c^4*d^5 + 10*a^2*b^8*c^3*d^6 - 10*a^3*b^7*c^2*d^7 + 5*a^4*b^6*c*d^8 - a^5*b^5*d^9)/(b
^2*c*d^9*abs(b) - a*b*d^10*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + 105/8*(b^6*c^3 - 3*a
*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*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^5*abs(b))