3.1492 \(\int \frac{(a+b x)^{7/2}}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=183 \[ -\frac{35 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 d^4}+\frac{35 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}{96 d^3}-\frac{7 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)}{24 d^2}+\frac{35 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 \sqrt{b} d^{9/2}}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 d} \]

[Out]

(-35*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*d^4) + (35*(b*c - a*d)^2*(a + b*x)^(3/2)*Sqrt[c + d*x])/(9
6*d^3) - (7*(b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*d^2) + ((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*d) + (35*
(b*c - a*d)^4*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*Sqrt[b]*d^(9/2))

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Rubi [A]  time = 0.0967502, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {50, 63, 217, 206} \[ -\frac{35 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 d^4}+\frac{35 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}{96 d^3}-\frac{7 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)}{24 d^2}+\frac{35 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 \sqrt{b} d^{9/2}}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(7/2)/Sqrt[c + d*x],x]

[Out]

(-35*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*d^4) + (35*(b*c - a*d)^2*(a + b*x)^(3/2)*Sqrt[c + d*x])/(9
6*d^3) - (7*(b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*d^2) + ((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*d) + (35*
(b*c - a*d)^4*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*Sqrt[b]*d^(9/2))

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)^{7/2}}{\sqrt{c+d x}} \, dx &=\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 d}-\frac{(7 (b c-a d)) \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx}{8 d}\\ &=-\frac{7 (b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 d}+\frac{\left (35 (b c-a d)^2\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{48 d^2}\\ &=\frac{35 (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}{96 d^3}-\frac{7 (b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 d}-\frac{\left (35 (b c-a d)^3\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{64 d^3}\\ &=-\frac{35 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}{64 d^4}+\frac{35 (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}{96 d^3}-\frac{7 (b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 d}+\frac{\left (35 (b c-a d)^4\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 d^4}\\ &=-\frac{35 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}{64 d^4}+\frac{35 (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}{96 d^3}-\frac{7 (b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 d}+\frac{\left (35 (b c-a d)^4\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 )}{64 b d^4}\\ &=-\frac{35 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}{64 d^4}+\frac{35 (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}{96 d^3}-\frac{7 (b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 d}+\frac{\left (35 (b c-a d)^4\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 )}{64 b d^4}\\ &=-\frac{35 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}{64 d^4}+\frac{35 (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}{96 d^3}-\frac{7 (b c-a d) (a+b x)^{5/2} \sqrt{c+d x}}{24 d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 d}+\frac{35 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 \sqrt{b} d^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.650824, size = 189, normalized size = 1.03 \[ \frac{\sqrt{d} \sqrt{a+b x} (c+d x) \left (a^2 b d^2 (326 d x-511 c)+279 a^3 d^3+a b^2 d \left (385 c^2-252 c d x+200 d^2 x^2\right )+b^3 \left (70 c^2 d x-105 c^3-56 c d^2 x^2+48 d^3 x^3\right )\right )+\frac{105 (b c-a d)^{9/2} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{b}}{192 d^{9/2} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d]*Sqrt[a + b*x]*(c + d*x)*(279*a^3*d^3 + a^2*b*d^2*(-511*c + 326*d*x) + a*b^2*d*(385*c^2 - 252*c*d*x +
200*d^2*x^2) + b^3*(-105*c^3 + 70*c^2*d*x - 56*c*d^2*x^2 + 48*d^3*x^3)) + (105*(b*c - a*d)^(9/2)*Sqrt[(b*(c +
d*x))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/b)/(192*d^(9/2)*Sqrt[c + d*x])

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Maple [B]  time = 0.005, size = 650, normalized size = 3.6 \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)^(7/2)/(d*x+c)^(1/2),x)

[Out]

1/4*(b*x+a)^(7/2)*(d*x+c)^(1/2)/d+7/24/d*(b*x+a)^(5/2)*(d*x+c)^(1/2)*a-7/24/d^2*(b*x+a)^(5/2)*(d*x+c)^(1/2)*b*
c+35/96/d*(b*x+a)^(3/2)*(d*x+c)^(1/2)*a^2-35/48/d^2*(b*x+a)^(3/2)*(d*x+c)^(1/2)*a*b*c+35/96/d^3*(b*x+a)^(3/2)*
(d*x+c)^(1/2)*b^2*c^2+35/64/d*(b*x+a)^(1/2)*(d*x+c)^(1/2)*a^3-105/64/d^2*(b*x+a)^(1/2)*(d*x+c)^(1/2)*a^2*b*c+1
05/64/d^3*(b*x+a)^(1/2)*(d*x+c)^(1/2)*a*b^2*c^2-35/64/d^4*(b*x+a)^(1/2)*(d*x+c)^(1/2)*b^3*c^3+35/128*((b*x+a)*
(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1
/2))/(b*d)^(1/2)*a^4-35/32/d*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b
*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^3*b*c+105/64/d^2*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2
)/(d*x+c)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^2*b^2*c^
2-35/32/d^3*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*
b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a*b^3*c^3+35/128/d^4*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)
*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*b^4*c^4

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.96458, size = 1226, normalized size = 6.7 \begin{align*} \left [\frac{105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{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 x + b c + a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \,{\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 385 \, a b^{3} c^{2} d^{2} - 511 \, a^{2} b^{2} c d^{3} + 279 \, a^{3} b d^{4} - 8 \,{\left (7 \, b^{4} c d^{3} - 25 \, a b^{3} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{2} d^{2} - 126 \, a b^{3} c d^{3} + 163 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, b d^{5}}, -\frac{105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \,{\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 385 \, a b^{3} c^{2} d^{2} - 511 \, a^{2} b^{2} c d^{3} + 279 \, a^{3} b d^{4} - 8 \,{\left (7 \, b^{4} c d^{3} - 25 \, a b^{3} d^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{2} d^{2} - 126 \, a b^{3} c d^{3} + 163 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, b d^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(105*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*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*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d
+ a*b*d^2)*x) + 4*(48*b^4*d^4*x^3 - 105*b^4*c^3*d + 385*a*b^3*c^2*d^2 - 511*a^2*b^2*c*d^3 + 279*a^3*b*d^4 - 8
*(7*b^4*c*d^3 - 25*a*b^3*d^4)*x^2 + 2*(35*b^4*c^2*d^2 - 126*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sq
rt(d*x + c))/(b*d^5), -1/384*(105*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt
(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c
*d + a*b*d^2)*x)) - 2*(48*b^4*d^4*x^3 - 105*b^4*c^3*d + 385*a*b^3*c^2*d^2 - 511*a^2*b^2*c*d^3 + 279*a^3*b*d^4
- 8*(7*b^4*c*d^3 - 25*a*b^3*d^4)*x^2 + 2*(35*b^4*c^2*d^2 - 126*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x)*sqrt(b*x + a)
*sqrt(d*x + c))/(b*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)**(7/2)/(d*x+c)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.12165, size = 362, normalized size = 1.98 \begin{align*} \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b d} - \frac{7 \,{\left (b c d^{5} - a d^{6}\right )}}{b d^{7}}\right )} + \frac{35 \,{\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )}}{b d^{7}}\right )} - \frac{105 \,{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )}}{b d^{7}}\right )} \sqrt{b x + a} - \frac{105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\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^{4}}\right )} b}{192 \,{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/(b*d) - 7*(b*c*d^5 - a*d^6)/
(b*d^7)) + 35*(b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6)/(b*d^7)) - 105*(b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*
d^5 - a^3*d^6)/(b*d^7))*sqrt(b*x + a) - 105*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4
*d^4)*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^4))*b/abs(b)