3.1499 \(\int \frac{1}{(a+b x)^{7/2} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{16 d^2 \sqrt{c+d x}}{15 \sqrt{a+b x} (b c-a d)^3}+\frac{8 d \sqrt{c+d x}}{15 (a+b x)^{3/2} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{5 (a+b x)^{5/2} (b c-a d)} \]

[Out]

(-2*Sqrt[c + d*x])/(5*(b*c - a*d)*(a + b*x)^(5/2)) + (8*d*Sqrt[c + d*x])/(15*(b*c - a*d)^2*(a + b*x)^(3/2)) -
(16*d^2*Sqrt[c + d*x])/(15*(b*c - a*d)^3*Sqrt[a + b*x])

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Rubi [A]  time = 0.016225, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{16 d^2 \sqrt{c+d x}}{15 \sqrt{a+b x} (b c-a d)^3}+\frac{8 d \sqrt{c+d x}}{15 (a+b x)^{3/2} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{5 (a+b x)^{5/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(7/2)*Sqrt[c + d*x]),x]

[Out]

(-2*Sqrt[c + d*x])/(5*(b*c - a*d)*(a + b*x)^(5/2)) + (8*d*Sqrt[c + d*x])/(15*(b*c - a*d)^2*(a + b*x)^(3/2)) -
(16*d^2*Sqrt[c + d*x])/(15*(b*c - a*d)^3*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)^{7/2} \sqrt{c+d x}} \, dx &=-\frac{2 \sqrt{c+d x}}{5 (b c-a d) (a+b x)^{5/2}}-\frac{(4 d) \int \frac{1}{(a+b x)^{5/2} \sqrt{c+d x}} \, dx}{5 (b c-a d)}\\ &=-\frac{2 \sqrt{c+d x}}{5 (b c-a d) (a+b x)^{5/2}}+\frac{8 d \sqrt{c+d x}}{15 (b c-a d)^2 (a+b x)^{3/2}}+\frac{\left (8 d^2\right ) \int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx}{15 (b c-a d)^2}\\ &=-\frac{2 \sqrt{c+d x}}{5 (b c-a d) (a+b x)^{5/2}}+\frac{8 d \sqrt{c+d x}}{15 (b c-a d)^2 (a+b x)^{3/2}}-\frac{16 d^2 \sqrt{c+d x}}{15 (b c-a d)^3 \sqrt{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0304289, size = 75, normalized size = 0.74 \[ -\frac{2 \sqrt{c+d x} \left (15 a^2 d^2-10 a b d (c-2 d x)+b^2 \left (3 c^2-4 c d x+8 d^2 x^2\right )\right )}{15 (a+b x)^{5/2} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(7/2)*Sqrt[c + d*x]),x]

[Out]

(-2*Sqrt[c + d*x]*(15*a^2*d^2 - 10*a*b*d*(c - 2*d*x) + b^2*(3*c^2 - 4*c*d*x + 8*d^2*x^2)))/(15*(b*c - a*d)^3*(
a + b*x)^(5/2))

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Maple [A]  time = 0.005, size = 105, normalized size = 1. \begin{align*}{\frac{16\,{b}^{2}{d}^{2}{x}^{2}+40\,ab{d}^{2}x-8\,{b}^{2}cdx+30\,{a}^{2}{d}^{2}-20\,abcd+6\,{b}^{2}{c}^{2}}{15\,{a}^{3}{d}^{3}-45\,{a}^{2}bc{d}^{2}+45\,a{b}^{2}{c}^{2}d-15\,{b}^{3}{c}^{3}}\sqrt{dx+c} \left ( bx+a \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(7/2)/(d*x+c)^(1/2),x)

[Out]

2/15*(d*x+c)^(1/2)*(8*b^2*d^2*x^2+20*a*b*d^2*x-4*b^2*c*d*x+15*a^2*d^2-10*a*b*c*d+3*b^2*c^2)/(b*x+a)^(5/2)/(a^3
*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

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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)^(7/2)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.72415, size = 509, normalized size = 5.04 \begin{align*} -\frac{2 \,{\left (8 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2} - 4 \,{\left (b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{15 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} +{\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 )} x^{3} + 3 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/2)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

-2/15*(8*b^2*d^2*x^2 + 3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2 - 4*(b^2*c*d - 5*a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x
+ c)/(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3 + (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)*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*x^2 + 3*(a^2*b^4*c^3 - 3*a^
3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(7/2)/(d*x+c)**(1/2),x)

[Out]

Integral(1/((a + b*x)**(7/2)*sqrt(c + d*x)), x)

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Giac [B]  time = 1.14124, size = 306, normalized size = 3.03 \begin{align*} -\frac{32 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 5 \,{\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^{2} c + 5 \,{\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 d + 10 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} \sqrt{b d} b^{3} d^{2}}{15 \,{\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 )}^{5}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/2)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

-32/15*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 5*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^2*b^2*c + 5*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + 10*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)*sqrt(b*d)*b^3*d^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)^5*abs(b))