Optimal. Leaf size=221 \[ -\frac{d \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{7/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}+\frac{d \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}-\frac{d (c+d x)^{3/4}}{b \sqrt{a+b x} (b c-a d)}-\frac{2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.235718, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {47, 51, 63, 307, 224, 221, 1200, 1199, 424} \[ -\frac{d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}+\frac{d \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}-\frac{d (c+d x)^{3/4}}{b \sqrt{a+b x} (b c-a d)}-\frac{2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 51
Rule 63
Rule 307
Rule 224
Rule 221
Rule 1200
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx &=-\frac{2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}+\frac{d \int \frac{1}{(a+b x)^{3/2} \sqrt [4]{c+d x}} \, dx}{2 b}\\ &=-\frac{2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac{d (c+d x)^{3/4}}{b (b c-a d) \sqrt{a+b x}}+\frac{d^2 \int \frac{1}{\sqrt{a+b x} \sqrt [4]{c+d x}} \, dx}{4 b (b c-a d)}\\ &=-\frac{2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac{d (c+d x)^{3/4}}{b (b c-a d) \sqrt{a+b x}}+\frac{d \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{b (b c-a d)}\\ &=-\frac{2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac{d (c+d x)^{3/4}}{b (b c-a d) \sqrt{a+b x}}-\frac{d \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{b^{3/2} \sqrt{b c-a d}}+\frac{d \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{b^{3/2} \sqrt{b c-a d}}\\ &=-\frac{2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac{d (c+d x)^{3/4}}{b (b c-a d) \sqrt{a+b x}}-\frac{\left (d \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{b x^4}{\left (a-\frac{b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{b^{3/2} \sqrt{b c-a d} \sqrt{a+b x}}+\frac{\left (d \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}{\sqrt{1+\frac{b x^4}{\left (a-\frac{b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{b^{3/2} \sqrt{b c-a d} \sqrt{a+b x}}\\ &=-\frac{2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac{d (c+d x)^{3/4}}{b (b c-a d) \sqrt{a+b x}}-\frac{d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt [4]{b c-a d} \sqrt{a+b x}}+\frac{\left (d \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}}{\sqrt{1-\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{b^{3/2} \sqrt{b c-a d} \sqrt{a+b x}}\\ &=-\frac{2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac{d (c+d x)^{3/4}}{b (b c-a d) \sqrt{a+b x}}+\frac{d \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt [4]{b c-a d} \sqrt{a+b x}}-\frac{d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt [4]{b c-a d} \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.0239278, size = 73, normalized size = 0.33 \[ -\frac{2 (c+d x)^{3/4} \, _2F_1\left (-\frac{3}{2},-\frac{3}{4};-\frac{1}{2};\frac{d (a+b x)}{a d-b c}\right )}{3 b (a+b x)^{3/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx+c \right ) ^{{\frac{3}{4}}} \left ( bx+a \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{3}{4}}}{{\left (b x + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{4}}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x\right )^{\frac{3}{4}}}{\left (a + b x\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{3}{4}}}{{\left (b x + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]