3.1501 \(\int \frac{1}{(a+b x)^{11/2} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=171 \[ -\frac{256 d^4 \sqrt{c+d x}}{315 \sqrt{a+b x} (b c-a d)^5}+\frac{128 d^3 \sqrt{c+d x}}{315 (a+b x)^{3/2} (b c-a d)^4}-\frac{32 d^2 \sqrt{c+d x}}{105 (a+b x)^{5/2} (b c-a d)^3}+\frac{16 d \sqrt{c+d x}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{9 (a+b x)^{9/2} (b c-a d)} \]

[Out]

(-2*Sqrt[c + d*x])/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (16*d*Sqrt[c + d*x])/(63*(b*c - a*d)^2*(a + b*x)^(7/2)) -
(32*d^2*Sqrt[c + d*x])/(105*(b*c - a*d)^3*(a + b*x)^(5/2)) + (128*d^3*Sqrt[c + d*x])/(315*(b*c - a*d)^4*(a +
b*x)^(3/2)) - (256*d^4*Sqrt[c + d*x])/(315*(b*c - a*d)^5*Sqrt[a + b*x])

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Rubi [A]  time = 0.0409882, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{256 d^4 \sqrt{c+d x}}{315 \sqrt{a+b x} (b c-a d)^5}+\frac{128 d^3 \sqrt{c+d x}}{315 (a+b x)^{3/2} (b c-a d)^4}-\frac{32 d^2 \sqrt{c+d x}}{105 (a+b x)^{5/2} (b c-a d)^3}+\frac{16 d \sqrt{c+d x}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{9 (a+b x)^{9/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(11/2)*Sqrt[c + d*x]),x]

[Out]

(-2*Sqrt[c + d*x])/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (16*d*Sqrt[c + d*x])/(63*(b*c - a*d)^2*(a + b*x)^(7/2)) -
(32*d^2*Sqrt[c + d*x])/(105*(b*c - a*d)^3*(a + b*x)^(5/2)) + (128*d^3*Sqrt[c + d*x])/(315*(b*c - a*d)^4*(a +
b*x)^(3/2)) - (256*d^4*Sqrt[c + d*x])/(315*(b*c - a*d)^5*Sqrt[a + b*x])

Rule 45

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(a+b x)^{11/2} \sqrt{c+d x}} \, dx &=-\frac{2 \sqrt{c+d x}}{9 (b c-a d) (a+b x)^{9/2}}-\frac{(8 d) \int \frac{1}{(a+b x)^{9/2} \sqrt{c+d x}} \, dx}{9 (b c-a d)}\\ &=-\frac{2 \sqrt{c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac{16 d \sqrt{c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}+\frac{\left (16 d^2\right ) \int \frac{1}{(a+b x)^{7/2} \sqrt{c+d x}} \, dx}{21 (b c-a d)^2}\\ &=-\frac{2 \sqrt{c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac{16 d \sqrt{c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac{32 d^2 \sqrt{c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}-\frac{\left (64 d^3\right ) \int \frac{1}{(a+b x)^{5/2} \sqrt{c+d x}} \, dx}{105 (b c-a d)^3}\\ &=-\frac{2 \sqrt{c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac{16 d \sqrt{c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac{32 d^2 \sqrt{c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac{128 d^3 \sqrt{c+d x}}{315 (b c-a d)^4 (a+b x)^{3/2}}+\frac{\left (128 d^4\right ) \int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx}{315 (b c-a d)^4}\\ &=-\frac{2 \sqrt{c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac{16 d \sqrt{c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac{32 d^2 \sqrt{c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac{128 d^3 \sqrt{c+d x}}{315 (b c-a d)^4 (a+b x)^{3/2}}-\frac{256 d^4 \sqrt{c+d x}}{315 (b c-a d)^5 \sqrt{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0631599, size = 168, normalized size = 0.98 \[ -\frac{2 \sqrt{c+d x} \left (126 a^2 b^2 d^2 \left (3 c^2-4 c d x+8 d^2 x^2\right )-420 a^3 b d^3 (c-2 d x)+315 a^4 d^4+36 a b^3 d \left (6 c^2 d x-5 c^3-8 c d^2 x^2+16 d^3 x^3\right )+b^4 \left (48 c^2 d^2 x^2-40 c^3 d x+35 c^4-64 c d^3 x^3+128 d^4 x^4\right )\right )}{315 (a+b x)^{9/2} (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(11/2)*Sqrt[c + d*x]),x]

[Out]

(-2*Sqrt[c + d*x]*(315*a^4*d^4 - 420*a^3*b*d^3*(c - 2*d*x) + 126*a^2*b^2*d^2*(3*c^2 - 4*c*d*x + 8*d^2*x^2) + 3
6*a*b^3*d*(-5*c^3 + 6*c^2*d*x - 8*c*d^2*x^2 + 16*d^3*x^3) + b^4*(35*c^4 - 40*c^3*d*x + 48*c^2*d^2*x^2 - 64*c*d
^3*x^3 + 128*d^4*x^4)))/(315*(b*c - a*d)^5*(a + b*x)^(9/2))

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Maple [A]  time = 0.009, size = 256, normalized size = 1.5 \begin{align*}{\frac{256\,{b}^{4}{d}^{4}{x}^{4}+1152\,a{b}^{3}{d}^{4}{x}^{3}-128\,{b}^{4}c{d}^{3}{x}^{3}+2016\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-576\,a{b}^{3}c{d}^{3}{x}^{2}+96\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+1680\,{a}^{3}b{d}^{4}x-1008\,{a}^{2}{b}^{2}c{d}^{3}x+432\,a{b}^{3}{c}^{2}{d}^{2}x-80\,{b}^{4}{c}^{3}dx+630\,{a}^{4}{d}^{4}-840\,{a}^{3}bc{d}^{3}+756\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-360\,a{b}^{3}{c}^{3}d+70\,{b}^{4}{c}^{4}}{315\,{a}^{5}{d}^{5}-1575\,{a}^{4}bc{d}^{4}+3150\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-3150\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+1575\,a{b}^{4}{c}^{4}d-315\,{b}^{5}{c}^{5}}\sqrt{dx+c} \left ( bx+a \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(11/2)/(d*x+c)^(1/2),x)

[Out]

2/315*(d*x+c)^(1/2)*(128*b^4*d^4*x^4+576*a*b^3*d^4*x^3-64*b^4*c*d^3*x^3+1008*a^2*b^2*d^4*x^2-288*a*b^3*c*d^3*x
^2+48*b^4*c^2*d^2*x^2+840*a^3*b*d^4*x-504*a^2*b^2*c*d^3*x+216*a*b^3*c^2*d^2*x-40*b^4*c^3*d*x+315*a^4*d^4-420*a
^3*b*c*d^3+378*a^2*b^2*c^2*d^2-180*a*b^3*c^3*d+35*b^4*c^4)/(b*x+a)^(9/2)/(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2
*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)

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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(1/(b*x+a)^(11/2)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 48.8145, size = 1301, normalized size = 7.61 \begin{align*} -\frac{2 \,{\left (128 \, b^{4} d^{4} x^{4} + 35 \, b^{4} c^{4} - 180 \, a b^{3} c^{3} d + 378 \, a^{2} b^{2} c^{2} d^{2} - 420 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} - 64 \,{\left (b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 48 \,{\left (b^{4} c^{2} d^{2} - 6 \, a b^{3} c d^{3} + 21 \, a^{2} b^{2} d^{4}\right )} x^{2} - 8 \,{\left (5 \, b^{4} c^{3} d - 27 \, a b^{3} c^{2} d^{2} + 63 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{315 \,{\left (a^{5} b^{5} c^{5} - 5 \, a^{6} b^{4} c^{4} d + 10 \, a^{7} b^{3} c^{3} d^{2} - 10 \, a^{8} b^{2} c^{2} d^{3} + 5 \, a^{9} b c d^{4} - a^{10} d^{5} +{\left (b^{10} c^{5} - 5 \, a b^{9} c^{4} d + 10 \, a^{2} b^{8} c^{3} d^{2} - 10 \, a^{3} b^{7} c^{2} d^{3} + 5 \, a^{4} b^{6} c d^{4} - a^{5} b^{5} d^{5}\right )} x^{5} + 5 \,{\left (a b^{9} c^{5} - 5 \, a^{2} b^{8} c^{4} d + 10 \, a^{3} b^{7} c^{3} d^{2} - 10 \, a^{4} b^{6} c^{2} d^{3} + 5 \, a^{5} b^{5} c d^{4} - a^{6} b^{4} d^{5}\right )} x^{4} + 10 \,{\left (a^{2} b^{8} c^{5} - 5 \, a^{3} b^{7} c^{4} d + 10 \, a^{4} b^{6} c^{3} d^{2} - 10 \, a^{5} b^{5} c^{2} d^{3} + 5 \, a^{6} b^{4} c d^{4} - a^{7} b^{3} d^{5}\right )} x^{3} + 10 \,{\left (a^{3} b^{7} c^{5} - 5 \, a^{4} b^{6} c^{4} d + 10 \, a^{5} b^{5} c^{3} d^{2} - 10 \, a^{6} b^{4} c^{2} d^{3} + 5 \, a^{7} b^{3} c d^{4} - a^{8} b^{2} d^{5}\right )} x^{2} + 5 \,{\left (a^{4} b^{6} c^{5} - 5 \, a^{5} b^{5} c^{4} d + 10 \, a^{6} b^{4} c^{3} d^{2} - 10 \, a^{7} b^{3} c^{2} d^{3} + 5 \, a^{8} b^{2} c d^{4} - a^{9} b d^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(11/2)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

-2/315*(128*b^4*d^4*x^4 + 35*b^4*c^4 - 180*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 420*a^3*b*c*d^3 + 315*a^4*d^4 -
64*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 48*(b^4*c^2*d^2 - 6*a*b^3*c*d^3 + 21*a^2*b^2*d^4)*x^2 - 8*(5*b^4*c^3*d - 2
7*a*b^3*c^2*d^2 + 63*a^2*b^2*c*d^3 - 105*a^3*b*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^5*b^5*c^5 - 5*a^6*b^4*c^
4*d + 10*a^7*b^3*c^3*d^2 - 10*a^8*b^2*c^2*d^3 + 5*a^9*b*c*d^4 - a^10*d^5 + (b^10*c^5 - 5*a*b^9*c^4*d + 10*a^2*
b^8*c^3*d^2 - 10*a^3*b^7*c^2*d^3 + 5*a^4*b^6*c*d^4 - a^5*b^5*d^5)*x^5 + 5*(a*b^9*c^5 - 5*a^2*b^8*c^4*d + 10*a^
3*b^7*c^3*d^2 - 10*a^4*b^6*c^2*d^3 + 5*a^5*b^5*c*d^4 - a^6*b^4*d^5)*x^4 + 10*(a^2*b^8*c^5 - 5*a^3*b^7*c^4*d +
10*a^4*b^6*c^3*d^2 - 10*a^5*b^5*c^2*d^3 + 5*a^6*b^4*c*d^4 - a^7*b^3*d^5)*x^3 + 10*(a^3*b^7*c^5 - 5*a^4*b^6*c^4
*d + 10*a^5*b^5*c^3*d^2 - 10*a^6*b^4*c^2*d^3 + 5*a^7*b^3*c*d^4 - a^8*b^2*d^5)*x^2 + 5*(a^4*b^6*c^5 - 5*a^5*b^5
*c^4*d + 10*a^6*b^4*c^3*d^2 - 10*a^7*b^3*c^2*d^3 + 5*a^8*b^2*c*d^4 - a^9*b*d^5)*x)

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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(1/(b*x+a)**(11/2)/(d*x+c)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.26576, size = 805, normalized size = 4.71 \begin{align*} -\frac{512 \,{\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4} - 9 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{6} c^{3} + 27 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{2} d - 27 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c d^{2} + 9 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{3} d^{3} + 36 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{4} c^{2} - 72 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} c d + 36 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} d^{2} - 84 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{6} b^{2} c + 84 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{6} a b d + 126 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{8}\right )} \sqrt{b d} b^{5} d^{4}}{315 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{9}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(11/2)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

-512/315*(b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4 - 9*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^6*c^3 + 27*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^2*a*b^5*c^2*d - 27*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^4*c*d^
2 + 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^3*d^3 + 36*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^4*c^2 - 72*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^4*a*b^3*c*d + 36*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^2*d^2 - 8
4*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^2*c + 84*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b*d + 126*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^8)*sqrt(b*d)*b^5*d^4/((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)^9*a
bs(b))